Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 78.3%

Time: 20.8s

Alternatives: 14

Speedup: 0.8×

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: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-150}:\\ \;\;\;\;t_1 \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+241}\right):\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M) (* d 2.0)) 2.0))))))
        (t_1 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= t_0 -5e-150)
     (* t_1 (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* 0.5 M)) 2.0))) l)))
     (if (or (<= t_0 0.0) (not (<= t_0 1e+241)))
       (/ (fabs d) (sqrt (* h l)))
       (* (- 1.0 (* h (/ (pow (* D (/ 0.5 (/ d M))) 2.0) (/ l 0.5)))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M) / (d * 2.0)), 2.0))));
	double t_1 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (t_0 <= -5e-150) {
		tmp = t_1 * (1.0 - ((h * (0.5 * pow(((D / d) * (0.5 * M)), 2.0))) / l));
	} else if ((t_0 <= 0.0) || !(t_0 <= 1e+241)) {
		tmp = fabs(d) / sqrt((h * l));
	} else {
		tmp = (1.0 - (h * (pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5)))) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m) / (d * 2.0d0)) ** 2.0d0))))
    t_1 = sqrt((d / l)) * sqrt((d / h))
    if (t_0 <= (-5d-150)) then
        tmp = t_1 * (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (0.5d0 * m)) ** 2.0d0))) / l))
    else if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 1d+241))) then
        tmp = abs(d) / sqrt((h * l))
    else
        tmp = (1.0d0 - (h * (((d_1 * (0.5d0 / (d / m))) ** 2.0d0) / (l / 0.5d0)))) * t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M) / (d * 2.0)), 2.0))));
	double t_1 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double tmp;
	if (t_0 <= -5e-150) {
		tmp = t_1 * (1.0 - ((h * (0.5 * Math.pow(((D / d) * (0.5 * M)), 2.0))) / l));
	} else if ((t_0 <= 0.0) || !(t_0 <= 1e+241)) {
		tmp = Math.abs(d) / Math.sqrt((h * l));
	} else {
		tmp = (1.0 - (h * (Math.pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5)))) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M) / (d * 2.0)), 2.0))))
	t_1 = math.sqrt((d / l)) * math.sqrt((d / h))
	tmp = 0
	if t_0 <= -5e-150:
		tmp = t_1 * (1.0 - ((h * (0.5 * math.pow(((D / d) * (0.5 * M)), 2.0))) / l))
	elif (t_0 <= 0.0) or not (t_0 <= 1e+241):
		tmp = math.fabs(d) / math.sqrt((h * l))
	else:
		tmp = (1.0 - (h * (math.pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5)))) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (t_0 <= -5e-150)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0))) / l)));
	elseif ((t_0 <= 0.0) || !(t_0 <= 1e+241))
		tmp = Float64(abs(d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(0.5 / Float64(d / M))) ^ 2.0) / Float64(l / 0.5)))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D * M) / (d * 2.0)) ^ 2.0))));
	t_1 = sqrt((d / l)) * sqrt((d / h));
	tmp = 0.0;
	if (t_0 <= -5e-150)
		tmp = t_1 * (1.0 - ((h * (0.5 * (((D / d) * (0.5 * M)) ^ 2.0))) / l));
	elseif ((t_0 <= 0.0) || ~((t_0 <= 1e+241)))
		tmp = abs(d) / sqrt((h * l));
	else
		tmp = (1.0 - (h * (((D * (0.5 / (d / M))) ^ 2.0) / (l / 0.5)))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-150], N[(t$95$1 * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+241]], $MachinePrecision]], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(h * N[(N[Power[N[(D * N[(0.5 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4.9999999999999999e-150

   1. Initial program 90.9%

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

      1. metadata-eval 90.9%

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

      2. unpow1/2 90.9%

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

      3. metadata-eval 90.9%

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

      4. unpow1/2 90.9%

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

      5. *-commutative 90.9%

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

      6. associate-*l* 90.9%

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

      7. times-frac 90.9%

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

      8. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

      1. associate-*r* 90.9%

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

      2. frac-times 90.9%

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

      3. *-commutative 90.9%

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

      4. metadata-eval 90.9%

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

      5. associate-*r/ 90.9%

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

      6. metadata-eval 90.9%

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

      7. *-commutative 90.9%

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

      8. frac-times 90.9%

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

      9. *-commutative 90.9%

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

      10. div-inv 90.9%

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

      11. metadata-eval 90.9%

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

   5. Applied egg-rr 90.9%

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

   if -4.9999999999999999e-150 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -0.0 or 1.0000000000000001e241 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 27.1%

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

      1. associate-*l* 27.1%

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

      2. metadata-eval 27.1%

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

      3. unpow1/2 27.1%

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

      4. metadata-eval 27.1%

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

      5. unpow1/2 27.1%

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

      6. sub-neg 27.1%

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

      7. +-commutative 27.1%

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

      8. *-commutative 27.1%

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

      9. associate-*l* 27.1%

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

      10. distribute-rgt-neg-in 27.1%

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

      11. *-commutative 27.1%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in M around 0 34.5%

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

      1. *-rgt-identity 34.5%

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

      2. expm1-log1p-u 34.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 31.6%

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

      4. sqrt-unprod 29.3%

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

   6. Applied egg-rr 29.3%

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

      1. expm1-def 31.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 32.0%

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

   8. Simplified 32.0%

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

   9. Taylor expanded in d around 0 36.3%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. unpow2 36.3%

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

       2. *-commutative 36.3%

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

   11. Simplified 36.3%

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

       1. sqrt-div 42.4%

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

   13. Applied egg-rr 42.4%

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

       1. rem-sqrt-square 63.7%

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

   15. Simplified 63.7%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.0000000000000001e241

   1. Initial program 99.5%

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

      1. metadata-eval 99.5%

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

      2. unpow1/2 99.5%

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

      3. metadata-eval 99.5%

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

      4. unpow1/2 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*l* 99.5%

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

      7. times-frac 97.9%

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

      8. metadata-eval 97.9%

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

   3. Simplified 97.9%

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

      1. associate-*r* 97.9%

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

      2. frac-times 99.5%

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

      3. *-commutative 99.5%

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

      4. metadata-eval 99.5%

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

      5. clear-num 99.5%

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

      6. un-div-inv 99.5%

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

      7. metadata-eval 99.5%

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

      8. *-commutative 99.5%

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

      9. frac-times 97.9%

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

      10. *-commutative 97.9%

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

      11. div-inv 97.9%

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

      12. metadata-eval 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. associate-/r/ 97.9%

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

      2. *-commutative 97.9%

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

      3. associate-/l* 97.9%

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

      4. /-rgt-identity 97.9%

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

      5. associate-/l* 97.9%

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

      6. metadata-eval 97.9%

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

      7. times-frac 99.5%

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

      8. associate-*r/ 99.5%

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

      9. associate-/l/ 99.5%

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

      10. metadata-eval 99.5%

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

      11. associate-/l* 99.5%

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

      12. /-rgt-identity 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -5 \cdot 10^{-150}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{+241}\right):\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]

Alternative 2: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= d -1e-310)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
      (- 1.0 (* h (/ (pow (* D (/ 0.5 (/ d M))) 2.0) (/ l 0.5)))))
     (if (<= d 5.8e-224)
       (* (/ (* (* D M) (* D M)) d) (* (/ (sqrt h) (pow l 1.5)) -0.125))
       (*
        (* t_0 (/ (sqrt d) (sqrt h)))
        (- 1.0 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (* 0.5 (/ h l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (d <= -1e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (h * (pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5))));
	} else if (d <= 5.8e-224) {
		tmp = (((D * M) * (D * M)) / d) * ((sqrt(h) / pow(l, 1.5)) * -0.125);
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * (h / 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 = sqrt((d / l))
    if (d <= (-1d-310)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0d0 - (h * (((d_1 * (0.5d0 / (d / m))) ** 2.0d0) / (l / 0.5d0))))
    else if (d <= 5.8d-224) then
        tmp = (((d_1 * m) * (d_1 * m)) / d) * ((sqrt(h) / (l ** 1.5d0)) * (-0.125d0))
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (0.5d0 * (h / 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 tmp;
	if (d <= -1e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * (1.0 - (h * (Math.pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5))));
	} else if (d <= 5.8e-224) {
		tmp = (((D * M) * (D * M)) / d) * ((Math.sqrt(h) / Math.pow(l, 1.5)) * -0.125);
	} else {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - (Math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * (h / l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if d <= -1e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * (1.0 - (h * (math.pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5))))
	elif d <= 5.8e-224:
		tmp = (((D * M) * (D * M)) / d) * ((math.sqrt(h) / math.pow(l, 1.5)) * -0.125)
	else:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - (math.pow(((M / 2.0) * (D / d)), 2.0) * (0.5 * (h / l))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * Float64(1.0 - Float64(h * Float64((Float64(D * Float64(0.5 / Float64(d / M))) ^ 2.0) / Float64(l / 0.5)))));
	elseif (d <= 5.8e-224)
		tmp = Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) / d) * Float64(Float64(sqrt(h) / (l ^ 1.5)) * -0.125));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1e-310)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (h * (((D * (0.5 / (d / M))) ^ 2.0) / (l / 0.5))));
	elseif (d <= 5.8e-224)
		tmp = (((D * M) * (D * M)) / d) * ((sqrt(h) / (l ^ 1.5)) * -0.125);
	else
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - ((((M / 2.0) * (D / d)) ^ 2.0) * (0.5 * (h / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(h * N[(N[Power[N[(D * N[(0.5 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-224], N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.999999999999969e-311

   1. Initial program 65.9%

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

      1. metadata-eval 65.9%

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

      2. unpow1/2 65.9%

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

      3. metadata-eval 65.9%

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

      4. unpow1/2 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-*l* 65.9%

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

      7. times-frac 65.9%

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

      8. metadata-eval 65.9%

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

   3. Simplified 65.9%

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

      1. associate-*r* 65.9%

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

      2. frac-times 65.9%

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

      3. *-commutative 65.9%

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

      4. metadata-eval 65.9%

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

      5. clear-num 65.9%

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

      6. un-div-inv 66.8%

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

      7. metadata-eval 66.8%

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

      8. *-commutative 66.8%

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

      9. frac-times 66.8%

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

      10. *-commutative 66.8%

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

      11. div-inv 66.8%

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

      12. metadata-eval 66.8%

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

   5. Applied egg-rr 66.8%

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

      1. associate-/r/ 69.5%

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

      2. *-commutative 69.5%

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

      3. associate-/l* 69.5%

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

      4. /-rgt-identity 69.5%

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

      5. associate-/l* 69.5%

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

      6. metadata-eval 69.5%

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

      7. times-frac 69.4%

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

      8. associate-*r/ 69.4%

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

      9. associate-/l/ 69.4%

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

      10. metadata-eval 69.4%

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

      11. associate-/l* 69.4%

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

      12. /-rgt-identity 69.4%

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

      13. *-commutative 69.4%

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

      14. associate-/l* 69.4%

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

   7. Simplified 69.4%

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

      1. frac-2neg 69.4%

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

      2. sqrt-div 80.6%

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

   9. Applied egg-rr 80.6%

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

   if -9.999999999999969e-311 < d < 5.8000000000000001e-224

   1. Initial program 34.0%

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

      1. metadata-eval 34.0%

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

      2. unpow1/2 34.0%

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

      3. metadata-eval 34.0%

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

      4. unpow1/2 34.0%

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

      5. *-commutative 34.0%

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

      6. associate-*l* 34.0%

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

      7. times-frac 22.9%

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

      8. metadata-eval 22.9%

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

   3. Simplified 22.9%

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

      1. associate-*r* 22.9%

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

      2. frac-times 34.0%

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

      3. *-commutative 34.0%

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

      4. metadata-eval 34.0%

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

      5. clear-num 34.0%

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

      6. un-div-inv 34.0%

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

      7. metadata-eval 34.0%

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

      8. *-commutative 34.0%

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

      9. frac-times 22.9%

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

      10. *-commutative 22.9%

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

      11. div-inv 22.9%

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

      12. metadata-eval 22.9%

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

   5. Applied egg-rr 22.9%

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

      1. associate-/r/ 23.2%

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

      2. *-commutative 23.2%

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

      3. associate-/l* 23.2%

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

      4. /-rgt-identity 23.2%

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

      5. associate-/l* 23.2%

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

      6. metadata-eval 23.2%

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

      7. times-frac 34.2%

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

      8. associate-*r/ 34.2%

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

      9. associate-/l/ 34.2%

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

      10. metadata-eval 34.2%

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

      11. associate-/l* 34.2%

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

      12. /-rgt-identity 34.2%

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

      13. *-commutative 34.2%

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

      14. associate-/l* 34.2%

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

   7. Simplified 34.2%

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

   8. Taylor expanded in d around 0 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-*l* 28.6%

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

      3. unpow2 28.6%

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

      4. unpow2 28.6%

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

      5. unswap-sqr 45.4%

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

   10. Simplified 45.4%

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

       1. sqrt-div 45.4%

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

   12. Applied egg-rr 45.4%

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

       1. sqr-pow 45.4%

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

       2. rem-sqrt-square 72.2%

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

       3. sqr-pow 72.2%

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

       4. fabs-sqr 72.2%

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

       5. sqr-pow 72.2%

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

       6. metadata-eval 72.2%

          \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right) \]

   14. Simplified 72.2%

       \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right) \]

   if 5.8000000000000001e-224 < d

   1. Initial program 70.2%

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

      1. metadata-eval 70.2%

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

      2. unpow1/2 70.2%

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

      3. metadata-eval 70.2%

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

      4. unpow1/2 70.2%

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

      5. *-commutative 70.2%

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

      6. associate-*l* 70.2%

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

      7. times-frac 71.0%

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

      8. metadata-eval 71.0%

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

   3. Simplified 71.0%

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

      1. sqrt-div 80.3%

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

      2. div-inv 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. associate-*r/ 80.3%

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

      2. *-rgt-identity 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 3: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 7.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= D 7.2e+46)
   (/ (fabs d) (sqrt (* h l)))
   (*
    (- 1.0 (* h (/ (pow (* D (/ 0.5 (/ d M))) 2.0) (/ l 0.5))))
    (* (sqrt (/ d l)) (sqrt (/ d h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 7.2e+46) {
		tmp = fabs(d) / sqrt((h * l));
	} else {
		tmp = (1.0 - (h * (pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5)))) * (sqrt((d / l)) * sqrt((d / 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 (d_1 <= 7.2d+46) then
        tmp = abs(d) / sqrt((h * l))
    else
        tmp = (1.0d0 - (h * (((d_1 * (0.5d0 / (d / m))) ** 2.0d0) / (l / 0.5d0)))) * (sqrt((d / l)) * sqrt((d / 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 (D <= 7.2e+46) {
		tmp = Math.abs(d) / Math.sqrt((h * l));
	} else {
		tmp = (1.0 - (h * (Math.pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5)))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if D <= 7.2e+46:
		tmp = math.fabs(d) / math.sqrt((h * l))
	else:
		tmp = (1.0 - (h * (math.pow((D * (0.5 / (d / M))), 2.0) / (l / 0.5)))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 7.2e+46)
		tmp = Float64(abs(d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(0.5 / Float64(d / M))) ^ 2.0) / Float64(l / 0.5)))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 7.2e+46)
		tmp = abs(d) / sqrt((h * l));
	else
		tmp = (1.0 - (h * (((D * (0.5 / (d / M))) ^ 2.0) / (l / 0.5)))) * (sqrt((d / l)) * sqrt((d / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[D, 7.2e+46], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(h * N[(N[Power[N[(D * N[(0.5 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 7.1999999999999997e46

   1. Initial program 65.2%

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

      1. associate-*l* 64.7%

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

      2. metadata-eval 64.7%

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

      3. unpow1/2 64.7%

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

      4. metadata-eval 64.7%

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

      5. unpow1/2 64.7%

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

      6. sub-neg 64.7%

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

      7. +-commutative 64.7%

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

      8. *-commutative 64.7%

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

      9. associate-*l* 64.7%

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

      10. distribute-rgt-neg-in 64.7%

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

      11. *-commutative 64.7%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in M around 0 45.2%

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

      1. *-rgt-identity 45.2%

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

      2. expm1-log1p-u 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 29.6%

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

      4. sqrt-unprod 25.9%

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

   6. Applied egg-rr 25.9%

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

      1. expm1-def 37.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 38.4%

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

   8. Simplified 38.4%

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

   9. Taylor expanded in d around 0 32.7%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. unpow2 32.7%

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

       2. *-commutative 32.7%

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

   11. Simplified 32.7%

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

       1. sqrt-div 39.2%

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

   13. Applied egg-rr 39.2%

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

       1. rem-sqrt-square 53.9%

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

   15. Simplified 53.9%

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

   if 7.1999999999999997e46 < D

   1. Initial program 67.2%

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

      1. metadata-eval 67.2%

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

      2. unpow1/2 67.2%

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

      3. metadata-eval 67.2%

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

      4. unpow1/2 67.2%

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

      5. *-commutative 67.2%

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

      6. associate-*l* 67.2%

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

      7. times-frac 67.1%

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

      8. metadata-eval 67.1%

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

   3. Simplified 67.1%

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

      1. associate-*r* 67.1%

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

      2. frac-times 67.2%

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

      3. *-commutative 67.2%

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

      4. metadata-eval 67.2%

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

      5. clear-num 65.3%

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

      6. un-div-inv 65.3%

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

      7. metadata-eval 65.3%

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

      8. *-commutative 65.3%

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

      9. frac-times 65.2%

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

      10. *-commutative 65.2%

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

      11. div-inv 65.2%

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

      12. metadata-eval 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. associate-/r/ 67.1%

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

      2. *-commutative 67.1%

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

      3. associate-/l* 67.1%

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

      4. /-rgt-identity 67.1%

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

      5. associate-/l* 67.1%

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

      6. metadata-eval 67.1%

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

      7. times-frac 67.2%

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

      8. associate-*r/ 65.3%

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

      9. associate-/l/ 65.3%

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

      10. metadata-eval 65.3%

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

      11. associate-/l* 65.3%

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

      12. /-rgt-identity 65.3%

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

      13. *-commutative 65.3%

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

      14. associate-/l* 65.3%

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

   7. Simplified 65.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]

Alternative 4: 51.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;M \leq 6.4 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 4.4 \cdot 10^{-267}:\\ \;\;\;\;t_0 \cdot \left(1 - h \cdot \left(\frac{D \cdot M}{\frac{\ell}{D \cdot M}} \cdot \frac{0.125}{d \cdot d}\right)\right)\\ \mathbf{elif}\;M \leq 7 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(0.125 \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ d l)) (sqrt (/ d h))))
        (t_1 (/ (fabs d) (sqrt (* h l)))))
   (if (<= M 6.4e-284)
     t_1
     (if (<= M 4.4e-267)
       (* t_0 (- 1.0 (* h (* (/ (* D M) (/ l (* D M))) (/ 0.125 (* d d))))))
       (if (<= M 7e-247)
         t_1
         (if (<= M 1.35e-220)
           (*
            (sqrt (* (/ d l) (/ d h)))
            (+ 1.0 (* (* h (/ (pow (* 0.5 (/ (* D M) d)) 2.0) l)) -0.5)))
           (if (<= M 3.2e-140)
             t_1
             (*
              t_0
              (-
               1.0
               (* (* (/ D d) (/ D d)) (* 0.125 (/ (* M (* h M)) l))))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l)) * sqrt((d / h));
	double t_1 = fabs(d) / sqrt((h * l));
	double tmp;
	if (M <= 6.4e-284) {
		tmp = t_1;
	} else if (M <= 4.4e-267) {
		tmp = t_0 * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))));
	} else if (M <= 7e-247) {
		tmp = t_1;
	} else if (M <= 1.35e-220) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((h * (pow((0.5 * ((D * M) / d)), 2.0) / l)) * -0.5));
	} else if (M <= 3.2e-140) {
		tmp = t_1;
	} else {
		tmp = t_0 * (1.0 - (((D / d) * (D / d)) * (0.125 * ((M * (h * M)) / 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 / l)) * sqrt((d / h))
    t_1 = abs(d) / sqrt((h * l))
    if (m <= 6.4d-284) then
        tmp = t_1
    else if (m <= 4.4d-267) then
        tmp = t_0 * (1.0d0 - (h * (((d_1 * m) / (l / (d_1 * m))) * (0.125d0 / (d * d)))))
    else if (m <= 7d-247) then
        tmp = t_1
    else if (m <= 1.35d-220) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((h * (((0.5d0 * ((d_1 * m) / d)) ** 2.0d0) / l)) * (-0.5d0)))
    else if (m <= 3.2d-140) then
        tmp = t_1
    else
        tmp = t_0 * (1.0d0 - (((d_1 / d) * (d_1 / d)) * (0.125d0 * ((m * (h * m)) / 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)) * Math.sqrt((d / h));
	double t_1 = Math.abs(d) / Math.sqrt((h * l));
	double tmp;
	if (M <= 6.4e-284) {
		tmp = t_1;
	} else if (M <= 4.4e-267) {
		tmp = t_0 * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))));
	} else if (M <= 7e-247) {
		tmp = t_1;
	} else if (M <= 1.35e-220) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + ((h * (Math.pow((0.5 * ((D * M) / d)), 2.0) / l)) * -0.5));
	} else if (M <= 3.2e-140) {
		tmp = t_1;
	} else {
		tmp = t_0 * (1.0 - (((D / d) * (D / d)) * (0.125 * ((M * (h * M)) / l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l)) * math.sqrt((d / h))
	t_1 = math.fabs(d) / math.sqrt((h * l))
	tmp = 0
	if M <= 6.4e-284:
		tmp = t_1
	elif M <= 4.4e-267:
		tmp = t_0 * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))))
	elif M <= 7e-247:
		tmp = t_1
	elif M <= 1.35e-220:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + ((h * (math.pow((0.5 * ((D * M) / d)), 2.0) / l)) * -0.5))
	elif M <= 3.2e-140:
		tmp = t_1
	else:
		tmp = t_0 * (1.0 - (((D / d) * (D / d)) * (0.125 * ((M * (h * M)) / l))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	t_1 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (M <= 6.4e-284)
		tmp = t_1;
	elseif (M <= 4.4e-267)
		tmp = Float64(t_0 * Float64(1.0 - Float64(h * Float64(Float64(Float64(D * M) / Float64(l / Float64(D * M))) * Float64(0.125 / Float64(d * d))))));
	elseif (M <= 7e-247)
		tmp = t_1;
	elseif (M <= 1.35e-220)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(Float64(h * Float64((Float64(0.5 * Float64(Float64(D * M) / d)) ^ 2.0) / l)) * -0.5)));
	elseif (M <= 3.2e-140)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(0.125 * Float64(Float64(M * Float64(h * M)) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l)) * sqrt((d / h));
	t_1 = abs(d) / sqrt((h * l));
	tmp = 0.0;
	if (M <= 6.4e-284)
		tmp = t_1;
	elseif (M <= 4.4e-267)
		tmp = t_0 * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))));
	elseif (M <= 7e-247)
		tmp = t_1;
	elseif (M <= 1.35e-220)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((h * (((0.5 * ((D * M) / d)) ^ 2.0) / l)) * -0.5));
	elseif (M <= 3.2e-140)
		tmp = t_1;
	else
		tmp = t_0 * (1.0 - (((D / d) * (D / d)) * (0.125 * ((M * (h * M)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 6.4e-284], t$95$1, If[LessEqual[M, 4.4e-267], N[(t$95$0 * N[(1.0 - N[(h * N[(N[(N[(D * M), $MachinePrecision] / N[(l / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 7e-247], t$95$1, If[LessEqual[M, 1.35e-220], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(0.5 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 3.2e-140], t$95$1, N[(t$95$0 * N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < 6.40000000000000047e-284 or 4.39999999999999976e-267 < M < 6.9999999999999998e-247 or 1.35e-220 < M < 3.2000000000000001e-140

   1. Initial program 63.0%

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

      1. associate-*l* 62.9%

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

      2. metadata-eval 62.9%

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

      3. unpow1/2 62.9%

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

      4. metadata-eval 62.9%

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

      5. unpow1/2 62.9%

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

      6. sub-neg 62.9%

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

      7. +-commutative 62.9%

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

      8. *-commutative 62.9%

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

      9. associate-*l* 62.9%

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

      10. distribute-rgt-neg-in 62.9%

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

      11. *-commutative 62.9%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in M around 0 40.3%

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

      1. *-rgt-identity 40.3%

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

      2. expm1-log1p-u 38.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 28.6%

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

      4. sqrt-unprod 24.4%

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

   6. Applied egg-rr 24.4%

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

      1. expm1-def 33.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 34.2%

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

   8. Simplified 34.2%

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

   9. Taylor expanded in d around 0 30.0%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. unpow2 30.0%

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

       2. *-commutative 30.0%

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

   11. Simplified 30.0%

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

       1. sqrt-div 37.2%

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

   13. Applied egg-rr 37.2%

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

       1. rem-sqrt-square 52.4%

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

   15. Simplified 52.4%

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

   if 6.40000000000000047e-284 < M < 4.39999999999999976e-267

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

      2. unpow1/2 100.0%

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

      3. metadata-eval 100.0%

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

      4. unpow1/2 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. times-frac 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r* 100.0%

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

      2. frac-times 100.0%

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

      3. *-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. clear-num 100.0%

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

      6. un-div-inv 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. frac-times 100.0%

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

      10. *-commutative 100.0%

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

      11. div-inv 100.0%

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

      12. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-/r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. times-frac 100.0%

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

      8. associate-*r/ 100.0%

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

      9. associate-/l/ 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/l* 100.0%

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

      12. /-rgt-identity 100.0%

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

      13. *-commutative 100.0%

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

      14. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in D around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. times-frac 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. unswap-sqr 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. *-commutative 100.0%

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

      11. unpow2 100.0%

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

   10. Simplified 100.0%

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

   if 6.9999999999999998e-247 < M < 1.35e-220

   1. Initial program 79.7%

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

      1. metadata-eval 79.7%

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

      2. unpow1/2 79.7%

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

      3. metadata-eval 79.7%

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

      4. unpow1/2 79.7%

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

      5. *-commutative 79.7%

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

      6. associate-*l* 79.7%

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

      7. times-frac 79.4%

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

      8. metadata-eval 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r* 79.4%

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

      2. frac-times 79.7%

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

      3. *-commutative 79.7%

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

      4. metadata-eval 79.7%

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

      5. clear-num 79.7%

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

      6. un-div-inv 79.7%

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

      7. metadata-eval 79.7%

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

      8. *-commutative 79.7%

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

      9. frac-times 79.2%

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

      10. *-commutative 79.2%

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

      11. div-inv 79.2%

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

      12. metadata-eval 79.2%

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

   5. Applied egg-rr 79.2%

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

      1. associate-/r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-/l* 80.2%

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

      4. /-rgt-identity 80.2%

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

      5. associate-/l* 80.2%

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

      6. metadata-eval 80.2%

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

      7. times-frac 80.6%

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

      8. associate-*r/ 80.2%

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

      9. associate-/l/ 80.2%

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

      10. metadata-eval 80.2%

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

      11. associate-/l* 80.2%

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

      12. /-rgt-identity 80.2%

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

      13. *-commutative 80.2%

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

      14. associate-/l* 80.2%

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

   7. Simplified 80.2%

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

      1. expm1-log1p-u 38.9%

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

      2. expm1-udef 19.7%

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

      3. sqrt-unprod 19.7%

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

      4. associate-/r/ 19.7%

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

      5. associate-/r/ 19.7%

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

   9. Applied egg-rr 19.7%

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

   11. Simplified 80.6%

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

   if 3.2000000000000001e-140 < M

   1. Initial program 69.1%

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

      1. metadata-eval 69.1%

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

      2. unpow1/2 69.1%

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

      3. metadata-eval 69.1%

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

      4. unpow1/2 69.1%

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

      5. *-commutative 69.1%

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

      6. associate-*l* 69.1%

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

      7. times-frac 69.2%

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

      8. metadata-eval 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in M around 0 48.8%

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

      1. *-commutative 48.8%

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

      2. *-commutative 48.8%

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

      3. times-frac 47.6%

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

      4. *-commutative 47.6%

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

      5. associate-*l* 47.6%

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

      6. unpow2 47.6%

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

      7. unpow2 47.6%

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

      8. times-frac 62.1%

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

      9. unpow2 62.1%

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

      10. associate-*r* 62.2%

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

   6. Simplified 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{\ell} \cdot 0.125\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.4 \cdot 10^{-284}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;M \leq 4.4 \cdot 10^{-267}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \left(\frac{D \cdot M}{\frac{\ell}{D \cdot M}} \cdot \frac{0.125}{d \cdot d}\right)\right)\\ \mathbf{elif}\;M \leq 7 \cdot 10^{-247}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(0.125 \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \end{array} \]

Alternative 5: 66.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \left(\frac{D \cdot M}{\frac{\ell}{D \cdot M}} \cdot \frac{0.125}{d \cdot d}\right)\right)\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{-106}:\\ \;\;\;\;\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (* h (* (/ (* D M) (/ l (* D M))) (/ 0.125 (* d d))))))))
   (if (<= d -1.45e+169)
     (* d (- (sqrt (/ (/ 1.0 l) h))))
     (if (<= d -1e-309)
       t_0
       (if (<= d 1.08e-106)
         (* (/ (* (* D M) (* D M)) d) (* (/ (sqrt h) (pow l 1.5)) -0.125))
         (if (<= d 5.9e+119) t_0 (* d (sqrt (/ 1.0 (* h l))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))));
	double tmp;
	if (d <= -1.45e+169) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (d <= -1e-309) {
		tmp = t_0;
	} else if (d <= 1.08e-106) {
		tmp = (((D * M) * (D * M)) / d) * ((sqrt(h) / pow(l, 1.5)) * -0.125);
	} else if (d <= 5.9e+119) {
		tmp = t_0;
	} else {
		tmp = d * sqrt((1.0 / (h * 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 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (h * (((d_1 * m) / (l / (d_1 * m))) * (0.125d0 / (d * d)))))
    if (d <= (-1.45d+169)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else if (d <= (-1d-309)) then
        tmp = t_0
    else if (d <= 1.08d-106) then
        tmp = (((d_1 * m) * (d_1 * m)) / d) * ((sqrt(h) / (l ** 1.5d0)) * (-0.125d0))
    else if (d <= 5.9d+119) then
        tmp = t_0
    else
        tmp = d * sqrt((1.0d0 / (h * 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)) * Math.sqrt((d / h))) * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))));
	double tmp;
	if (d <= -1.45e+169) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (d <= -1e-309) {
		tmp = t_0;
	} else if (d <= 1.08e-106) {
		tmp = (((D * M) * (D * M)) / d) * ((Math.sqrt(h) / Math.pow(l, 1.5)) * -0.125);
	} else if (d <= 5.9e+119) {
		tmp = t_0;
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))))
	tmp = 0
	if d <= -1.45e+169:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif d <= -1e-309:
		tmp = t_0
	elif d <= 1.08e-106:
		tmp = (((D * M) * (D * M)) / d) * ((math.sqrt(h) / math.pow(l, 1.5)) * -0.125)
	elif d <= 5.9e+119:
		tmp = t_0
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(h * Float64(Float64(Float64(D * M) / Float64(l / Float64(D * M))) * Float64(0.125 / Float64(d * d))))))
	tmp = 0.0
	if (d <= -1.45e+169)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (d <= -1e-309)
		tmp = t_0;
	elseif (d <= 1.08e-106)
		tmp = Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) / d) * Float64(Float64(sqrt(h) / (l ^ 1.5)) * -0.125));
	elseif (d <= 5.9e+119)
		tmp = t_0;
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (h * (((D * M) / (l / (D * M))) * (0.125 / (d * d)))));
	tmp = 0.0;
	if (d <= -1.45e+169)
		tmp = d * -sqrt(((1.0 / l) / h));
	elseif (d <= -1e-309)
		tmp = t_0;
	elseif (d <= 1.08e-106)
		tmp = (((D * M) * (D * M)) / d) * ((sqrt(h) / (l ^ 1.5)) * -0.125);
	elseif (d <= 5.9e+119)
		tmp = t_0;
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(D * M), $MachinePrecision] / N[(l / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.45e+169], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -1e-309], t$95$0, If[LessEqual[d, 1.08e-106], N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.9e+119], t$95$0, N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.45e169

   1. Initial program 68.7%

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

      1. metadata-eval 68.7%

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

      2. unpow1/2 68.7%

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

      3. metadata-eval 68.7%

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

      4. unpow1/2 68.7%

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

      5. *-commutative 68.7%

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

      6. associate-*l* 68.7%

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

      7. times-frac 68.9%

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

      8. metadata-eval 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in M around 0 39.4%

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

      1. *-commutative 39.4%

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

      2. *-commutative 39.4%

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

      3. times-frac 36.0%

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

      4. *-commutative 36.0%

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

      5. associate-*l* 36.0%

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

      6. unpow2 36.0%

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

      7. unpow2 36.0%

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

      8. times-frac 65.5%

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

      9. unpow2 65.5%

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

      10. associate-*r* 65.8%

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

   6. Simplified 65.8%

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

      1. clear-num 65.8%

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

      2. sqrt-div 68.6%

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

      3. metadata-eval 68.6%

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

   8. Applied egg-rr 68.6%

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

   9. Taylor expanded in d around -inf 80.6%

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

       1. mul-1-neg 80.6%

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

       2. *-commutative 80.6%

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

       3. distribute-rgt-neg-in 80.6%

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

       4. associate-/r* 81.2%

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

   11. Simplified 81.2%

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

   if -1.45e169 < d < -1.000000000000002e-309 or 1.08e-106 < d < 5.9000000000000001e119

   1. Initial program 70.7%

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

      1. metadata-eval 70.7%

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

      2. unpow1/2 70.7%

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

      3. metadata-eval 70.7%

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

      4. unpow1/2 70.7%

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

      5. *-commutative 70.7%

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

      6. associate-*l* 70.7%

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

      7. times-frac 70.7%

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

      8. metadata-eval 70.7%

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

   3. Simplified 70.7%

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

      1. associate-*r* 70.7%

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

      2. frac-times 70.7%

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

      3. *-commutative 70.7%

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

      4. metadata-eval 70.7%

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

      5. clear-num 69.9%

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

      6. un-div-inv 70.6%

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

      7. metadata-eval 70.6%

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

      8. *-commutative 70.6%

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

      9. frac-times 70.6%

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

      10. *-commutative 70.6%

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

      11. div-inv 70.6%

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

      12. metadata-eval 70.6%

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

   5. Applied egg-rr 70.6%

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

      1. associate-/r/ 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-/l* 74.4%

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

      4. /-rgt-identity 74.4%

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

      5. associate-/l* 74.4%

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

      6. metadata-eval 74.4%

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

      7. times-frac 74.4%

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

      8. associate-*r/ 73.7%

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

      9. associate-/l/ 73.7%

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

      10. metadata-eval 73.7%

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

      11. associate-/l* 73.7%

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

      12. /-rgt-identity 73.7%

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

      13. *-commutative 73.7%

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

      14. associate-/l* 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in D around 0 58.3%

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

      1. associate-*r/ 58.3%

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

      2. *-commutative 58.3%

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

      3. times-frac 60.0%

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

      4. *-commutative 60.0%

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

      5. unpow2 60.0%

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

      6. unpow2 60.0%

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

      7. unswap-sqr 68.9%

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

      8. associate-/l* 72.1%

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

      9. *-commutative 72.1%

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

      10. *-commutative 72.1%

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

      11. unpow2 72.1%

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

   10. Simplified 72.1%

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

   if -1.000000000000002e-309 < d < 1.08e-106

   1. Initial program 49.9%

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

      1. metadata-eval 49.9%

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

      2. unpow1/2 49.9%

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

      3. metadata-eval 49.9%

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

      4. unpow1/2 49.9%

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

      5. *-commutative 49.9%

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

      6. associate-*l* 49.9%

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

      7. times-frac 45.6%

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

      8. metadata-eval 45.6%

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

   3. Simplified 45.6%

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

      1. associate-*r* 45.6%

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

      2. frac-times 49.9%

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

      3. *-commutative 49.9%

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

      4. metadata-eval 49.9%

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

      5. clear-num 49.9%

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

      6. un-div-inv 49.9%

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

      7. metadata-eval 49.9%

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

      8. *-commutative 49.9%

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

      9. frac-times 45.6%

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

      10. *-commutative 45.6%

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

      11. div-inv 45.6%

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

      12. metadata-eval 45.6%

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

   5. Applied egg-rr 45.6%

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

      1. associate-/r/ 45.7%

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

      2. *-commutative 45.7%

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

      3. associate-/l* 45.7%

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

      4. /-rgt-identity 45.7%

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

      5. associate-/l* 45.7%

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

      6. metadata-eval 45.7%

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

      7. times-frac 50.0%

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

      8. associate-*r/ 49.9%

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

      9. associate-/l/ 49.9%

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

      10. metadata-eval 49.9%

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

      11. associate-/l* 49.9%

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

      12. /-rgt-identity 49.9%

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

      13. *-commutative 49.9%

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

      14. associate-/l* 49.9%

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

   7. Simplified 49.9%

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

   8. Taylor expanded in d around 0 30.8%

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

      1. *-commutative 30.8%

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

      2. associate-*l* 30.8%

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

      3. unpow2 30.8%

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

      4. unpow2 30.8%

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

      5. unswap-sqr 48.0%

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

   10. Simplified 48.0%

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

       1. sqrt-div 50.1%

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

   12. Applied egg-rr 50.1%

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

       1. sqr-pow 50.1%

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

       2. rem-sqrt-square 66.6%

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

       3. sqr-pow 66.6%

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

       4. fabs-sqr 66.6%

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

       5. sqr-pow 66.6%

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

       6. metadata-eval 66.6%

          \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot -0.125\right) \]

   14. Simplified 66.6%

       \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot -0.125\right) \]

   if 5.9000000000000001e119 < d

   1. Initial program 64.0%

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

      1. metadata-eval 64.0%

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

      2. unpow1/2 64.0%

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

      3. metadata-eval 64.0%

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

      4. unpow1/2 64.0%

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

      5. *-commutative 64.0%

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

      6. associate-*l* 64.0%

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

      7. times-frac 66.5%

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

      8. metadata-eval 66.5%

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

   3. Simplified 66.5%

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

      1. associate-*r* 66.5%

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

      2. frac-times 64.0%

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

      3. *-commutative 64.0%

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

      4. metadata-eval 64.0%

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

      5. clear-num 64.0%

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

      6. un-div-inv 64.0%

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

      7. metadata-eval 64.0%

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

      8. *-commutative 64.0%

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

      9. frac-times 66.5%

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

      10. *-commutative 66.5%

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

      11. div-inv 66.5%

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

      12. metadata-eval 66.5%

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

   5. Applied egg-rr 66.5%

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

      1. associate-/r/ 69.3%

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

      2. *-commutative 69.3%

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

      3. associate-/l* 69.3%

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

      4. /-rgt-identity 69.3%

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

      5. associate-/l* 69.3%

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

      6. metadata-eval 69.3%

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

      7. times-frac 66.8%

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

      8. associate-*r/ 69.3%

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

      9. associate-/l/ 69.3%

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

      10. metadata-eval 69.3%

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

      11. associate-/l* 69.3%

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

      12. /-rgt-identity 69.3%

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

      13. *-commutative 69.3%

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

      14. associate-/l* 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in d around inf 70.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \left(\frac{D \cdot M}{\frac{\ell}{D \cdot M}} \cdot \frac{0.125}{d \cdot d}\right)\right)\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{-106}:\\ \;\;\;\;\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \left(\frac{D \cdot M}{\frac{\ell}{D \cdot M}} \cdot \frac{0.125}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 6: 50.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 7.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= D 7.5e+49)
   (/ (fabs d) (sqrt (* h l)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* (* h (/ (pow (* 0.5 (/ (* D M) d)) 2.0) l)) -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 7.5e+49) {
		tmp = fabs(d) / sqrt((h * l));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((h * (pow((0.5 * ((D * M) / d)), 2.0) / l)) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 7.5d+49) then
        tmp = abs(d) / sqrt((h * l))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((h * (((0.5d0 * ((d_1 * m) / d)) ** 2.0d0) / l)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 7.5e+49) {
		tmp = Math.abs(d) / Math.sqrt((h * l));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + ((h * (Math.pow((0.5 * ((D * M) / d)), 2.0) / l)) * -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if D <= 7.5e+49:
		tmp = math.fabs(d) / math.sqrt((h * l))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + ((h * (math.pow((0.5 * ((D * M) / d)), 2.0) / l)) * -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 7.5e+49)
		tmp = Float64(abs(d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(Float64(h * Float64((Float64(0.5 * Float64(Float64(D * M) / d)) ^ 2.0) / l)) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 7.5e+49)
		tmp = abs(d) / sqrt((h * l));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + ((h * (((0.5 * ((D * M) / d)) ^ 2.0) / l)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[D, 7.5e+49], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(0.5 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 7.4999999999999995e49

   1. Initial program 65.5%

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

      1. associate-*l* 65.0%

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

      2. metadata-eval 65.0%

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

      3. unpow1/2 65.0%

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

      4. metadata-eval 65.0%

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

      5. unpow1/2 65.0%

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

      6. sub-neg 65.0%

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

      7. +-commutative 65.0%

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

      8. *-commutative 65.0%

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

      9. associate-*l* 65.0%

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

      10. distribute-rgt-neg-in 65.0%

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

      11. *-commutative 65.0%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in M around 0 45.1%

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

      1. *-rgt-identity 45.1%

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

      2. expm1-log1p-u 43.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 29.4%

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

      4. sqrt-unprod 25.7%

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

   6. Applied egg-rr 25.7%

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

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 38.1%

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

   8. Simplified 38.1%

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

   9. Taylor expanded in d around 0 32.4%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. unpow2 32.4%

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

       2. *-commutative 32.4%

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

   11. Simplified 32.4%

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

       1. sqrt-div 39.1%

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

   13. Applied egg-rr 39.1%

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

       1. rem-sqrt-square 53.7%

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

   15. Simplified 53.7%

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

   if 7.4999999999999995e49 < D

   1. Initial program 65.9%

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

      1. metadata-eval 65.9%

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

      2. unpow1/2 65.9%

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

      3. metadata-eval 65.9%

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

      4. unpow1/2 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-*l* 65.9%

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

      7. times-frac 65.8%

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

      8. metadata-eval 65.8%

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

   3. Simplified 65.8%

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

      1. associate-*r* 65.8%

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

      2. frac-times 65.9%

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

      3. *-commutative 65.9%

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

      4. metadata-eval 65.9%

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

      5. clear-num 63.9%

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

      6. un-div-inv 63.9%

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

      7. metadata-eval 63.9%

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

      8. *-commutative 63.9%

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

      9. frac-times 63.8%

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

      10. *-commutative 63.8%

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

      11. div-inv 63.8%

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

      12. metadata-eval 63.8%

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

   5. Applied egg-rr 63.8%

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

      1. associate-/r/ 65.8%

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

      2. *-commutative 65.8%

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

      3. associate-/l* 65.8%

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

      4. /-rgt-identity 65.8%

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

      5. associate-/l* 65.8%

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

      6. metadata-eval 65.8%

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

      7. times-frac 65.9%

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

      8. associate-*r/ 63.9%

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

      9. associate-/l/ 63.9%

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

      10. metadata-eval 63.9%

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

      11. associate-/l* 63.9%

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

      12. /-rgt-identity 63.9%

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

      13. *-commutative 63.9%

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

      14. associate-/l* 63.9%

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

   7. Simplified 63.9%

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

      1. expm1-log1p-u 12.0%

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

      2. expm1-udef 10.1%

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

      3. sqrt-unprod 8.3%

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

      4. associate-/r/ 8.3%

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

      5. associate-/r/ 8.3%

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

   9. Applied egg-rr 8.3%

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

   11. Simplified 53.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 7: 45.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 3.8e-36)
   (/ (fabs d) (sqrt (* h l)))
   (*
    (sqrt (/ (/ d h) (/ l d)))
    (+ 1.0 (* -0.125 (* (/ (* D D) (* d d)) (/ (* M M) (/ l h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 3.8e-36) {
		tmp = fabs(d) / sqrt((h * l));
	} else {
		tmp = sqrt(((d / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (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 (m <= 3.8d-36) then
        tmp = abs(d) / sqrt((h * l))
    else
        tmp = sqrt(((d / h) / (l / d))) * (1.0d0 + ((-0.125d0) * (((d_1 * d_1) / (d * d)) * ((m * m) / (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 (M <= 3.8e-36) {
		tmp = Math.abs(d) / Math.sqrt((h * l));
	} else {
		tmp = Math.sqrt(((d / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (l / h)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 3.8e-36:
		tmp = math.fabs(d) / math.sqrt((h * l))
	else:
		tmp = math.sqrt(((d / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (l / h)))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 3.8e-36)
		tmp = Float64(abs(d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) / Float64(l / d))) * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(M * M) / Float64(l / h))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 3.8e-36)
		tmp = abs(d) / sqrt((h * l));
	else
		tmp = sqrt(((d / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (l / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 3.8e-36], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.79999999999999971e-36

   1. Initial program 64.1%

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

      1. associate-*l* 63.6%

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

      2. metadata-eval 63.6%

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

      3. unpow1/2 63.6%

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

      4. metadata-eval 63.6%

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

      5. unpow1/2 63.6%

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

      6. sub-neg 63.6%

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

      7. +-commutative 63.6%

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

      8. *-commutative 63.6%

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

      9. associate-*l* 63.6%

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

      10. distribute-rgt-neg-in 63.6%

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

      11. *-commutative 63.6%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in M around 0 42.6%

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

      1. *-rgt-identity 42.6%

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

      2. expm1-log1p-u 41.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 27.9%

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

      4. sqrt-unprod 24.3%

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

   6. Applied egg-rr 24.3%

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

      1. expm1-def 34.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in d around 0 30.7%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. unpow2 30.7%

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

       2. *-commutative 30.7%

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

   11. Simplified 30.7%

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

       1. sqrt-div 37.4%

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

   13. Applied egg-rr 37.4%

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

       1. rem-sqrt-square 52.0%

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

   15. Simplified 52.0%

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

   if 3.79999999999999971e-36 < M

   1. Initial program 69.7%

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

      1. metadata-eval 69.7%

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

      2. unpow1/2 69.7%

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

      3. metadata-eval 69.7%

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

      4. unpow1/2 69.7%

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

      5. *-commutative 69.7%

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

      6. associate-*l* 69.7%

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

      7. times-frac 69.7%

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

      8. metadata-eval 69.7%

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

   3. Simplified 69.7%

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

      1. sqrt-div 41.3%

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

      2. div-inv 41.3%

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

   5. Applied egg-rr 41.3%

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

      1. associate-*r/ 41.3%

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

      2. *-rgt-identity 41.3%

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

   7. Simplified 41.3%

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

      1. pow1 41.3%

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

      2. sqrt-div 69.7%

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

      3. sqrt-prod 58.2%

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

      4. frac-times 48.1%

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

      5. *-commutative 48.1%

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

      6. div-inv 48.1%

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

      7. metadata-eval 48.1%

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

   9. Applied egg-rr 48.1%

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

       1. unpow1 48.1%

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

       2. times-frac 58.2%

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

       3. associate-*r/ 51.0%

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

       4. associate-/l* 58.2%

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

       5. sub-neg 58.2%

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

       6. +-commutative 58.2%

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

       7. distribute-rgt-neg-in 58.2%

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

       8. fma-def 58.2%

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

   11. Simplified 58.2%

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

   12. Taylor expanded in D around 0 35.8%

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

       1. +-commutative 35.8%

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

       2. fma-def 35.8%

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

       3. times-frac 35.6%

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

       4. unpow2 35.6%

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

       5. unpow2 35.6%

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

       6. times-frac 48.9%

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

       7. unpow2 48.9%

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

       8. unpow2 48.9%

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

       9. associate-*r* 49.0%

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

       10. associate-/l* 50.5%

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

       11. fma-def 50.5%

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

       12. *-commutative 50.5%

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

       13. associate-*r* 50.5%

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

       14. +-commutative 50.5%

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

       15. associate-*r* 50.5%

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

       16. *-commutative 50.5%

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

   14. Simplified 35.6%

       \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 8: 44.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ t_1 := d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ (/ d h) (/ l d)))
          (+ 1.0 (* -0.125 (* (/ (* D D) (* d d)) (/ (* M M) (/ l h)))))))
        (t_1 (* d (- (sqrt (/ (/ 1.0 l) h))))))
   (if (<= l -5.2e+96)
     t_1
     (if (<= l -4.8e-40)
       t_0
       (if (<= l -3.3e-245)
         t_1
         (if (<= l 1.5e-108) t_0 (* d (sqrt (/ 1.0 (* h l))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (l / h)))));
	double t_1 = d * -sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -5.2e+96) {
		tmp = t_1;
	} else if (l <= -4.8e-40) {
		tmp = t_0;
	} else if (l <= -3.3e-245) {
		tmp = t_1;
	} else if (l <= 1.5e-108) {
		tmp = t_0;
	} else {
		tmp = d * sqrt((1.0 / (h * 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 / h) / (l / d))) * (1.0d0 + ((-0.125d0) * (((d_1 * d_1) / (d * d)) * ((m * m) / (l / h)))))
    t_1 = d * -sqrt(((1.0d0 / l) / h))
    if (l <= (-5.2d+96)) then
        tmp = t_1
    else if (l <= (-4.8d-40)) then
        tmp = t_0
    else if (l <= (-3.3d-245)) then
        tmp = t_1
    else if (l <= 1.5d-108) then
        tmp = t_0
    else
        tmp = d * sqrt((1.0d0 / (h * 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 / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (l / h)))));
	double t_1 = d * -Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -5.2e+96) {
		tmp = t_1;
	} else if (l <= -4.8e-40) {
		tmp = t_0;
	} else if (l <= -3.3e-245) {
		tmp = t_1;
	} else if (l <= 1.5e-108) {
		tmp = t_0;
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (l / h)))))
	t_1 = d * -math.sqrt(((1.0 / l) / h))
	tmp = 0
	if l <= -5.2e+96:
		tmp = t_1
	elif l <= -4.8e-40:
		tmp = t_0
	elif l <= -3.3e-245:
		tmp = t_1
	elif l <= 1.5e-108:
		tmp = t_0
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(Float64(d / h) / Float64(l / d))) * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(M * M) / Float64(l / h))))))
	t_1 = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))))
	tmp = 0.0
	if (l <= -5.2e+96)
		tmp = t_1;
	elseif (l <= -4.8e-40)
		tmp = t_0;
	elseif (l <= -3.3e-245)
		tmp = t_1;
	elseif (l <= 1.5e-108)
		tmp = t_0;
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) / (l / d))) * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((M * M) / (l / h)))));
	t_1 = d * -sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= -5.2e+96)
		tmp = t_1;
	elseif (l <= -4.8e-40)
		tmp = t_0;
	elseif (l <= -3.3e-245)
		tmp = t_1;
	elseif (l <= 1.5e-108)
		tmp = t_0;
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[l, -5.2e+96], t$95$1, If[LessEqual[l, -4.8e-40], t$95$0, If[LessEqual[l, -3.3e-245], t$95$1, If[LessEqual[l, 1.5e-108], t$95$0, N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.2e96 or -4.79999999999999982e-40 < l < -3.3000000000000001e-245

   1. Initial program 60.3%

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

      1. metadata-eval 60.3%

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

      2. unpow1/2 60.3%

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

      3. metadata-eval 60.3%

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

      4. unpow1/2 60.3%

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

      5. *-commutative 60.3%

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

      6. associate-*l* 60.3%

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

      7. times-frac 60.4%

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

      8. metadata-eval 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in M around 0 38.5%

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

      1. *-commutative 38.5%

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

      2. *-commutative 38.5%

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

      3. times-frac 39.7%

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

      4. *-commutative 39.7%

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

      5. associate-*l* 39.7%

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

      6. unpow2 39.7%

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

      7. unpow2 39.7%

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

      8. times-frac 53.1%

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

      9. unpow2 53.1%

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

      10. associate-*r* 53.3%

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

   6. Simplified 53.3%

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

      1. clear-num 52.1%

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

      2. sqrt-div 53.7%

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

      3. metadata-eval 53.7%

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

   8. Applied egg-rr 53.7%

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

   9. Taylor expanded in d around -inf 58.3%

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

       1. mul-1-neg 58.3%

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

       2. *-commutative 58.3%

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

       3. distribute-rgt-neg-in 58.3%

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

       4. associate-/r* 58.6%

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

   11. Simplified 58.6%

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

   if -5.2e96 < l < -4.79999999999999982e-40 or -3.3000000000000001e-245 < l < 1.49999999999999996e-108

   1. Initial program 76.0%

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

      1. metadata-eval 76.0%

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

      2. unpow1/2 76.0%

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

      3. metadata-eval 76.0%

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

      4. unpow1/2 76.0%

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

      5. *-commutative 76.0%

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

      6. associate-*l* 76.0%

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

      7. times-frac 76.0%

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

      8. metadata-eval 76.0%

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

   3. Simplified 76.0%

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

      1. sqrt-div 42.0%

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

      2. div-inv 42.0%

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

   5. Applied egg-rr 42.0%

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

      1. associate-*r/ 42.0%

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

      2. *-rgt-identity 42.0%

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

   7. Simplified 42.0%

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

      1. pow1 42.0%

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

      2. sqrt-div 76.0%

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

      3. sqrt-prod 69.2%

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

      4. frac-times 55.3%

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

      5. *-commutative 55.3%

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

      6. div-inv 55.3%

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

      7. metadata-eval 55.3%

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

   9. Applied egg-rr 55.3%

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

       1. unpow1 55.3%

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

       2. times-frac 69.2%

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

       3. associate-*r/ 63.2%

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

       4. associate-/l* 69.2%

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

       5. sub-neg 69.2%

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

       6. +-commutative 69.2%

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

       7. distribute-rgt-neg-in 69.2%

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

       8. fma-def 69.2%

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

   11. Simplified 68.1%

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

   12. Taylor expanded in D around 0 53.3%

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

       1. +-commutative 53.3%

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

       2. fma-def 53.3%

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

       3. times-frac 55.6%

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

       4. unpow2 55.6%

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

       5. unpow2 55.6%

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

       6. times-frac 65.1%

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

       7. unpow2 65.1%

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

       8. unpow2 65.1%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \mathsf{fma}\left(-0.125, {\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}, 1\right) \]

       9. associate-*r* 65.8%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \mathsf{fma}\left(-0.125, {\left(\frac{D}{d}\right)}^{2} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}, 1\right) \]

       10. associate-/l* 68.1%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \mathsf{fma}\left(-0.125, {\left(\frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}, 1\right) \]

       11. fma-def 68.1%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \color{blue}{\left(-0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right) + 1\right)} \]

       12. *-commutative 68.1%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(\color{blue}{\left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right) \cdot -0.125} + 1\right) \]

       13. associate-*r* 68.1%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(\color{blue}{{\left(\frac{D}{d}\right)}^{2} \cdot \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot -0.125\right)} + 1\right) \]

       14. +-commutative 68.1%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \color{blue}{\left(1 + {\left(\frac{D}{d}\right)}^{2} \cdot \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot -0.125\right)\right)} \]

       15. associate-*r* 68.1%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 + \color{blue}{\left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right) \cdot -0.125}\right) \]

       16. *-commutative 68.1%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 + \color{blue}{-0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right)}\right) \]

   14. Simplified 51.7%

       \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \color{blue}{\left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)} \]

   if 1.49999999999999996e-108 < l

   1. Initial program 60.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 60.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 60.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      6. un-div-inv 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      7. metadata-eval 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      8. *-commutative 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5}}{\frac{\ell}{h}}\right) \]

      9. frac-times 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      10. *-commutative 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      11. div-inv 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      12. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

   5. Applied egg-rr 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell} \cdot h}\right) \]

      2. *-commutative 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell}}\right) \]

      3. associate-/l* 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

      4. /-rgt-identity 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      5. associate-/l* 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      6. metadata-eval 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      7. times-frac 61.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      8. associate-*r/ 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      9. associate-/l/ 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      10. metadata-eval 62.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\frac{M}{\color{blue}{\frac{1}{0.5}}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      11. associate-/l* 62.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{\frac{M \cdot 0.5}{1}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      12. /-rgt-identity 62.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      13. *-commutative 62.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{0.5 \cdot M}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      14. associate-/l* 62.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{0.5}{\frac{d}{M}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

   7. Simplified 62.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

   8. Taylor expanded in d around inf 48.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+96}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{-245}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 9: 42.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq 8 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d 8e-167) (* (- d) t_0) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= 8e-167) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (d <= 8d-167) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= 8e-167) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if d <= 8e-167:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= 8e-167)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (d <= 8e-167)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 8e-167], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 8.00000000000000002e-167

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.1%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 62.1%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 62.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 62.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 62.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 62.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 62.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 62.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right) \]

      9. associate-*l* 62.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]

      10. distribute-rgt-neg-in 62.1%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]

      11. *-commutative 62.1%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right) \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 35.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 35.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. expm1-log1p-u 34.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 23.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} - 1} \]

      4. sqrt-unprod 21.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

   6. Applied egg-rr 21.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 30.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   8. Simplified 30.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   9. Taylor expanded in d around -inf 44.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 44.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 44.3%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 44.3%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. *-commutative 44.3%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-d\right) \]

   11. Simplified 44.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 8.00000000000000002e-167 < d

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      6. un-div-inv 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      7. metadata-eval 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      8. *-commutative 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5}}{\frac{\ell}{h}}\right) \]

      9. frac-times 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      10. *-commutative 70.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      11. div-inv 70.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      12. metadata-eval 70.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

   5. Applied egg-rr 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell} \cdot h}\right) \]

      2. *-commutative 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell}}\right) \]

      3. associate-/l* 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

      4. /-rgt-identity 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      5. associate-/l* 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      6. metadata-eval 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      7. times-frac 73.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      8. associate-*r/ 73.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      9. associate-/l/ 73.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      10. metadata-eval 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\frac{M}{\color{blue}{\frac{1}{0.5}}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      11. associate-/l* 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{\frac{M \cdot 0.5}{1}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      12. /-rgt-identity 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      13. *-commutative 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{0.5 \cdot M}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      14. associate-/l* 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{0.5}{\frac{d}{M}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

   7. Simplified 73.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

   8. Taylor expanded in d around inf 52.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 10: 42.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 8 \cdot 10^{-167}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 8e-167)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (* d (sqrt (/ 1.0 (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 8e-167) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 8d-167) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 8e-167) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 8e-167:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 8e-167)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 8e-167)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 8e-167], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 8.00000000000000002e-167

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 62.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 62.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in M around 0 43.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   5. Step-by-step derivation

      1. *-commutative 43.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. *-commutative 43.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot 0.125\right) \]

      3. times-frac 44.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot 0.125\right) \]

      4. *-commutative 44.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right) \cdot 0.125\right) \]

      5. associate-*l* 44.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.125\right)}\right) \]

      6. unpow2 44.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.125\right)\right) \]

      7. unpow2 44.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.125\right)\right) \]

      8. times-frac 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.125\right)\right) \]

      9. unpow2 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell} \cdot 0.125\right)\right) \]

      10. associate-*r* 57.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell} \cdot 0.125\right)\right) \]

   6. Simplified 57.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{\ell} \cdot 0.125\right)}\right) \]
   7. Step-by-step derivation

      1. clear-num 56.5%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{\ell} \cdot 0.125\right)\right) \]

      2. sqrt-div 57.3%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{\ell} \cdot 0.125\right)\right) \]

      3. metadata-eval 57.3%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{\ell} \cdot 0.125\right)\right) \]

   8. Applied egg-rr 57.3%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{\ell} \cdot 0.125\right)\right) \]

   9. Taylor expanded in d around -inf 44.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 44.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 44.3%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 44.3%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. associate-/r* 44.4%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(-d\right) \]

   11. Simplified 44.4%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if 8.00000000000000002e-167 < d

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      6. un-div-inv 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      7. metadata-eval 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      8. *-commutative 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5}}{\frac{\ell}{h}}\right) \]

      9. frac-times 70.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      10. *-commutative 70.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      11. div-inv 70.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      12. metadata-eval 70.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

   5. Applied egg-rr 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell} \cdot h}\right) \]

      2. *-commutative 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell}}\right) \]

      3. associate-/l* 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

      4. /-rgt-identity 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      5. associate-/l* 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      6. metadata-eval 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      7. times-frac 73.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      8. associate-*r/ 73.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      9. associate-/l/ 73.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      10. metadata-eval 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\frac{M}{\color{blue}{\frac{1}{0.5}}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      11. associate-/l* 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{\frac{M \cdot 0.5}{1}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      12. /-rgt-identity 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      13. *-commutative 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{0.5 \cdot M}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      14. associate-/l* 73.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{0.5}{\frac{d}{M}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

   7. Simplified 73.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

   8. Taylor expanded in d around inf 52.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8 \cdot 10^{-167}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 11: 38.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.7 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 1.7e-284) (sqrt (* (/ d l) (/ d h))) (/ d (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.7e-284) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / sqrt((h * 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 <= 1.7d-284) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d / sqrt((h * 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 <= 1.7e-284) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 1.7e-284:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 1.7e-284)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 1.7e-284)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 1.7e-284], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.69999999999999996e-284

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.2%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 65.2%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 65.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right) \]

      9. associate-*l* 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]

      10. distribute-rgt-neg-in 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]

      11. *-commutative 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 43.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. expm1-log1p-u 42.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 29.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} - 1} \]

      4. sqrt-unprod 26.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

   6. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 36.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 37.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   8. Simplified 37.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   if 1.69999999999999996e-284 < h

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.2%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 65.2%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 65.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right) \]

      9. associate-*l* 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]

      10. distribute-rgt-neg-in 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]

      11. *-commutative 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right) \]

   3. Simplified 63.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 34.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 34.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. expm1-log1p-u 33.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 21.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} - 1} \]

      4. sqrt-unprod 18.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

   6. Applied egg-rr 18.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 27.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 28.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   8. Simplified 28.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   9. Taylor expanded in d around 0 23.5%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. unpow2 23.5%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \]

       2. *-commutative 23.5%

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \]

   11. Simplified 23.5%

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. sqrt-div 29.5%

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]

   13. Applied egg-rr 29.5%

       \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. rem-sqrt-square 42.2%

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \]

       2. unpow1 42.2%

          \[\leadsto \frac{\left|\color{blue}{{d}^{1}}\right|}{\sqrt{h \cdot \ell}} \]

       3. sqr-pow 42.1%

          \[\leadsto \frac{\left|\color{blue}{{d}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}}\right|}{\sqrt{h \cdot \ell}} \]

       4. fabs-sqr 42.1%

          \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}}}{\sqrt{h \cdot \ell}} \]

       5. sqr-pow 42.2%

          \[\leadsto \frac{\color{blue}{{d}^{1}}}{\sqrt{h \cdot \ell}} \]

       6. unpow1 42.2%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   15. Simplified 42.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.7 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 12: 38.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 5.8 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 5.8e-284) (sqrt (* (/ d l) (/ d h))) (* d (sqrt (/ (/ 1.0 l) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 5.8e-284) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * sqrt(((1.0 / 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 (h <= 5.8d-284) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * sqrt(((1.0d0 / 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 (h <= 5.8e-284) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 5.8e-284:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 5.8e-284)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 5.8e-284)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 5.8e-284], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.8000000000000002e-284

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.2%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 65.2%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 65.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right) \]

      9. associate-*l* 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]

      10. distribute-rgt-neg-in 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]

      11. *-commutative 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 43.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. expm1-log1p-u 42.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 29.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} - 1} \]

      4. sqrt-unprod 26.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

   6. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 36.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 37.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   8. Simplified 37.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   if 5.8000000000000002e-284 < h

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 65.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 65.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      6. un-div-inv 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      7. metadata-eval 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      8. *-commutative 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5}}{\frac{\ell}{h}}\right) \]

      9. frac-times 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      10. *-commutative 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      11. div-inv 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      12. metadata-eval 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

   5. Applied egg-rr 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell} \cdot h}\right) \]

      2. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell}}\right) \]

      3. associate-/l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

      4. /-rgt-identity 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      5. associate-/l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      6. metadata-eval 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      7. times-frac 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      8. associate-*r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      9. associate-/l/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      10. metadata-eval 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\frac{M}{\color{blue}{\frac{1}{0.5}}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      11. associate-/l* 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{\frac{M \cdot 0.5}{1}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      12. /-rgt-identity 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      13. *-commutative 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{0.5 \cdot M}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      14. associate-/l* 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{0.5}{\frac{d}{M}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

   7. Simplified 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

   8. Taylor expanded in d around inf 42.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

      1. *-commutative 42.3%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. associate-/r* 42.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   10. Simplified 42.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 5.8 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 13: 37.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 9.2e-281) (sqrt (* (/ d l) (/ d h))) (* d (sqrt (/ 1.0 (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 9.2e-281) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * sqrt((1.0 / (h * 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 <= 9.2d-281) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * sqrt((1.0d0 / (h * 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 <= 9.2e-281) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 9.2e-281:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 9.2e-281)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 9.2e-281)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 9.2e-281], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 9.19999999999999956e-281

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 65.2%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 65.2%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 65.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right) \]

      9. associate-*l* 65.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]

      10. distribute-rgt-neg-in 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]

      11. *-commutative 65.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 43.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 43.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. expm1-log1p-u 42.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

      3. expm1-udef 29.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} - 1} \]

      4. sqrt-unprod 26.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

   6. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 36.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 37.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   8. Simplified 37.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   if 9.19999999999999956e-281 < h

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 65.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 65.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      6. un-div-inv 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      7. metadata-eval 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      8. *-commutative 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5}}{\frac{\ell}{h}}\right) \]

      9. frac-times 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      10. *-commutative 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      11. div-inv 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      12. metadata-eval 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

   5. Applied egg-rr 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell} \cdot h}\right) \]

      2. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\ell}}\right) \]

      3. associate-/l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

      4. /-rgt-identity 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      5. associate-/l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      6. metadata-eval 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      7. times-frac 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      8. associate-*r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}}{\frac{\ell}{0.5}}\right) \]

      9. associate-/l/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      10. metadata-eval 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\frac{M}{\color{blue}{\frac{1}{0.5}}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      11. associate-/l* 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{\frac{M \cdot 0.5}{1}}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      12. /-rgt-identity 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      13. *-commutative 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \frac{\color{blue}{0.5 \cdot M}}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

      14. associate-/l* 66.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{0.5}{\frac{d}{M}}}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

   7. Simplified 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

   8. Taylor expanded in d around inf 42.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 14: 27.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((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 / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. associate-*l* 65.2%

      \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

   2. metadata-eval 65.2%

      \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

   3. unpow1/2 65.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

   4. metadata-eval 65.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. unpow1/2 65.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. sub-neg 65.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

   7. +-commutative 65.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

   8. *-commutative 65.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right) \]

   9. associate-*l* 65.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]

   10. distribute-rgt-neg-in 65.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]

   11. *-commutative 65.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right) \]

3. Simplified 64.4%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)} \]

4. Taylor expanded in M around 0 39.0%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
5. Step-by-step derivation

   1. *-rgt-identity 39.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   2. expm1-log1p-u 37.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

   3. expm1-udef 25.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} - 1} \]

   4. sqrt-unprod 22.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

6. Applied egg-rr 22.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 32.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

   2. expm1-log1p 33.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

8. Simplified 33.0%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

9. Taylor expanded in d around 0 27.7%

   \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
10. Step-by-step derivation

    1. unpow2 27.7%

       \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \]

    2. *-commutative 27.7%

       \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \]

11. Simplified 27.7%

    \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]
12. Step-by-step derivation

    1. sqrt-div 33.4%

       \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]

13. Applied egg-rr 33.4%

    \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]
14. Step-by-step derivation

    1. rem-sqrt-square 46.7%

       \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \]

    2. unpow1 46.7%

       \[\leadsto \frac{\left|\color{blue}{{d}^{1}}\right|}{\sqrt{h \cdot \ell}} \]

    3. sqr-pow 21.5%

       \[\leadsto \frac{\left|\color{blue}{{d}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}}\right|}{\sqrt{h \cdot \ell}} \]

    4. fabs-sqr 21.5%

       \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}}}{\sqrt{h \cdot \ell}} \]

    5. sqr-pow 25.8%

       \[\leadsto \frac{\color{blue}{{d}^{1}}}{\sqrt{h \cdot \ell}} \]

    6. unpow1 25.8%

       \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

15. Simplified 25.8%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

16. Final simplification 25.8%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

Reproduce

herbie shell --seed 2023252 
(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)))))