Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 77.2%

Time: 25.9s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+127}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    d
    (*
     (/ 1.0 (sqrt (* l h)))
     (+ (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* h 0.5)) l) -1.0)))
   (if (<= l 1.02e+127)
     (*
      (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l)))
      (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l)))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = d * ((1.0 / sqrt((l * h))) * (((pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1.02e+127) {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - ((h * (0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = d * ((1.0d0 / sqrt((l * h))) * (((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0)))
    else if (l <= 1.02d+127) then
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0))) / l))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = d * ((1.0 / Math.sqrt((l * h))) * (((Math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1.02e+127) {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.sqrt((d / l))) * (1.0 - ((h * (0.5 * Math.pow(((D / d) * (M * 0.5)), 2.0))) / l));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-310:
		tmp = d * ((1.0 / math.sqrt((l * h))) * (((math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0))
	elif l <= 1.02e+127:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * (1.0 - ((h * (0.5 * math.pow(((D / d) * (M * 0.5)), 2.0))) / l))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(Float64(l * h))) * Float64(Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(h * 0.5)) / l) + -1.0)));
	elseif (l <= 1.02e+127)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = d * ((1.0 / sqrt((l * h))) * (((((M * (0.5 * (D / d))) ^ 2.0) * (h * 0.5)) / l) + -1.0));
	elseif (l <= 1.02e+127)
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - ((h * (0.5 * (((D / d) * (M * 0.5)) ^ 2.0))) / l));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(d * N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.02e+127], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 60.0%

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

      1. metadata-eval 60.0%

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

      2. unpow1/2 60.0%

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

      3. metadata-eval 60.0%

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

      4. unpow1/2 60.0%

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

      5. *-commutative 60.0%

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

      6. associate-*l* 60.0%

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

      7. times-frac 60.6%

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

      8. metadata-eval 60.6%

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

   3. Simplified 60.6%

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

      1. associate-*r* 60.6%

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

      2. frac-times 60.0%

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

      3. *-commutative 60.0%

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

      4. metadata-eval 60.0%

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

      5. associate-*r/ 62.6%

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

      6. metadata-eval 62.6%

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

      7. *-commutative 62.6%

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

      8. frac-times 63.1%

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

      9. div-inv 63.1%

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

      10. metadata-eval 63.1%

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

   5. Applied egg-rr 63.1%

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

      1. pow1 63.1%

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

      2. sqrt-unprod 48.7%

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

      3. associate-*l* 48.7%

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

      4. associate-*l* 48.7%

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

   7. Applied egg-rr 48.7%

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

      1. unpow1 48.7%

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

      2. associate-/l* 45.7%

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

      3. *-commutative 45.7%

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

   9. Simplified 45.7%

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

   10. Taylor expanded in d around -inf 62.3%

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

       1. associate-*r* 62.3%

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

       2. mul-1-neg 62.3%

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

   12. Simplified 62.3%

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

       1. pow1 62.3%

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

       2. associate-*l* 62.6%

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

       3. sqrt-div 63.3%

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

       4. metadata-eval 63.3%

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

       5. associate-/r/ 72.1%

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

   14. Applied egg-rr 72.1%

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

       1. unpow1 72.1%

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

       2. *-commutative 72.1%

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

       3. associate-*l/ 72.2%

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

   16. Simplified 72.2%

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

   if -1.999999999999994e-310 < l < 1.02e127

   1. Initial program 73.0%

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

      1. metadata-eval 73.0%

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

      2. unpow1/2 73.0%

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

      3. metadata-eval 73.0%

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

      4. unpow1/2 73.0%

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

      5. *-commutative 73.0%

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

      6. associate-*l* 73.0%

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

      7. times-frac 73.0%

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

      8. metadata-eval 73.0%

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

   3. Simplified 73.0%

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

      1. associate-*r* 73.0%

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

      2. frac-times 73.0%

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

      3. *-commutative 73.0%

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

      4. metadata-eval 73.0%

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

      5. associate-*r/ 77.3%

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

      6. metadata-eval 77.3%

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

      7. *-commutative 77.3%

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

      8. frac-times 77.4%

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

      9. div-inv 77.4%

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

      10. metadata-eval 77.4%

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

   5. Applied egg-rr 77.4%

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

      1. sqrt-div 85.7%

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

   7. Applied egg-rr 85.7%

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

   if 1.02e127 < l

   1. Initial program 53.2%

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

      1. metadata-eval 53.2%

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

      2. unpow1/2 53.2%

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

      3. metadata-eval 53.2%

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

      4. unpow1/2 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-*l* 53.2%

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

      7. times-frac 53.4%

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

      8. metadata-eval 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in d around inf 68.8%

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

      1. *-un-lft-identity 68.8%

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

      2. *-commutative 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. *-lft-identity 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-/r* 68.8%

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

   8. Simplified 68.8%

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

      1. sqrt-div 78.1%

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+127}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 2: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+127}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    d
    (*
     (/ 1.0 (sqrt (* l h)))
     (+ (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* h 0.5)) l) -1.0)))
   (if (<= l 3.2e+127)
     (*
      (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l))
      (* (sqrt (/ d h)) (/ (sqrt d) (sqrt l))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = d * ((1.0 / sqrt((l * h))) * (((pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 3.2e+127) {
		tmp = (1.0 - ((h * (0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (sqrt((d / h)) * (sqrt(d) / sqrt(l)));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = d * ((1.0d0 / sqrt((l * h))) * (((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0)))
    else if (l <= 3.2d+127) then
        tmp = (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0))) / l)) * (sqrt((d / h)) * (sqrt(d) / sqrt(l)))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = d * ((1.0 / Math.sqrt((l * h))) * (((Math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 3.2e+127) {
		tmp = (1.0 - ((h * (0.5 * Math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (Math.sqrt((d / h)) * (Math.sqrt(d) / Math.sqrt(l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-310:
		tmp = d * ((1.0 / math.sqrt((l * h))) * (((math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0))
	elif l <= 3.2e+127:
		tmp = (1.0 - ((h * (0.5 * math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (math.sqrt((d / h)) * (math.sqrt(d) / math.sqrt(l)))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(Float64(l * h))) * Float64(Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(h * 0.5)) / l) + -1.0)));
	elseif (l <= 3.2e+127)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l)) * Float64(sqrt(Float64(d / h)) * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = d * ((1.0 / sqrt((l * h))) * (((((M * (0.5 * (D / d))) ^ 2.0) * (h * 0.5)) / l) + -1.0));
	elseif (l <= 3.2e+127)
		tmp = (1.0 - ((h * (0.5 * (((D / d) * (M * 0.5)) ^ 2.0))) / l)) * (sqrt((d / h)) * (sqrt(d) / sqrt(l)));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(d * N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e+127], N[(N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 60.0%

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

      1. metadata-eval 60.0%

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

      2. unpow1/2 60.0%

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

      3. metadata-eval 60.0%

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

      4. unpow1/2 60.0%

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

      5. *-commutative 60.0%

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

      6. associate-*l* 60.0%

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

      7. times-frac 60.6%

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

      8. metadata-eval 60.6%

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

   3. Simplified 60.6%

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

      1. associate-*r* 60.6%

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

      2. frac-times 60.0%

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

      3. *-commutative 60.0%

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

      4. metadata-eval 60.0%

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

      5. associate-*r/ 62.6%

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

      6. metadata-eval 62.6%

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

      7. *-commutative 62.6%

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

      8. frac-times 63.1%

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

      9. div-inv 63.1%

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

      10. metadata-eval 63.1%

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

   5. Applied egg-rr 63.1%

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

      1. pow1 63.1%

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

      2. sqrt-unprod 48.7%

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

      3. associate-*l* 48.7%

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

      4. associate-*l* 48.7%

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

   7. Applied egg-rr 48.7%

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

      1. unpow1 48.7%

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

      2. associate-/l* 45.7%

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

      3. *-commutative 45.7%

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

   9. Simplified 45.7%

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

   10. Taylor expanded in d around -inf 62.3%

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

       1. associate-*r* 62.3%

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

       2. mul-1-neg 62.3%

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

   12. Simplified 62.3%

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

       1. pow1 62.3%

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

       2. associate-*l* 62.6%

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

       3. sqrt-div 63.3%

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

       4. metadata-eval 63.3%

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

       5. associate-/r/ 72.1%

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

   14. Applied egg-rr 72.1%

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

       1. unpow1 72.1%

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

       2. *-commutative 72.1%

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

       3. associate-*l/ 72.2%

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

   16. Simplified 72.2%

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

   if -1.999999999999994e-310 < l < 3.19999999999999976e127

   1. Initial program 73.0%

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

      1. metadata-eval 73.0%

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

      2. unpow1/2 73.0%

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

      3. metadata-eval 73.0%

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

      4. unpow1/2 73.0%

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

      5. *-commutative 73.0%

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

      6. associate-*l* 73.0%

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

      7. times-frac 73.0%

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

      8. metadata-eval 73.0%

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

   3. Simplified 73.0%

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

      1. associate-*r* 73.0%

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

      2. frac-times 73.0%

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

      3. *-commutative 73.0%

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

      4. metadata-eval 73.0%

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

      5. associate-*r/ 77.3%

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

      6. metadata-eval 77.3%

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

      7. *-commutative 77.3%

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

      8. frac-times 77.4%

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

      9. div-inv 77.4%

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

      10. metadata-eval 77.4%

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

   5. Applied egg-rr 77.4%

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

      1. sqrt-div 79.3%

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

   7. Applied egg-rr 79.3%

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

   if 3.19999999999999976e127 < l

   1. Initial program 53.2%

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

      1. metadata-eval 53.2%

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

      2. unpow1/2 53.2%

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

      3. metadata-eval 53.2%

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

      4. unpow1/2 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-*l* 53.2%

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

      7. times-frac 53.4%

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

      8. metadata-eval 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in d around inf 68.8%

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

      1. *-un-lft-identity 68.8%

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

      2. *-commutative 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. *-lft-identity 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-/r* 68.8%

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

   8. Simplified 68.8%

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

      1. sqrt-div 78.1%

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+127}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 3: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.5e-296)
   (*
    d
    (*
     (/ 1.0 (sqrt (* l h)))
     (+ (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* h 0.5)) l) -1.0)))
   (if (<= l 1.55e+120)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0))))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.5e-296) {
		tmp = d * ((1.0 / sqrt((l * h))) * (((pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1.55e+120) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.5d-296)) then
        tmp = d * ((1.0d0 / sqrt((l * h))) * (((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0)))
    else if (l <= 1.55d+120) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.5e-296) {
		tmp = d * ((1.0 / Math.sqrt((l * h))) * (((Math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1.55e+120) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.5e-296:
		tmp = d * ((1.0 / math.sqrt((l * h))) * (((math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0))
	elif l <= 1.55e+120:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.5e-296)
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(Float64(l * h))) * Float64(Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(h * 0.5)) / l) + -1.0)));
	elseif (l <= 1.55e+120)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.5e-296)
		tmp = d * ((1.0 / sqrt((l * h))) * (((((M * (0.5 * (D / d))) ^ 2.0) * (h * 0.5)) / l) + -1.0));
	elseif (l <= 1.55e+120)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0)))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.5e-296], N[(d * N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+120], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.49999999999999963e-296

   1. Initial program 59.4%

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

      1. metadata-eval 59.4%

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

      2. unpow1/2 59.4%

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

      3. metadata-eval 59.4%

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

      4. unpow1/2 59.4%

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

      5. *-commutative 59.4%

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

      6. associate-*l* 59.4%

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

      7. times-frac 60.0%

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

      8. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

      1. associate-*r* 60.0%

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

      2. frac-times 59.4%

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

      3. *-commutative 59.4%

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

      4. metadata-eval 59.4%

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

      5. associate-*r/ 62.0%

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

      6. metadata-eval 62.0%

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

      7. *-commutative 62.0%

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

      8. frac-times 62.5%

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

      9. div-inv 62.5%

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

      10. metadata-eval 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. pow1 62.5%

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

      2. sqrt-unprod 47.9%

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

      3. associate-*l* 47.9%

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

      4. associate-*l* 47.9%

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

   7. Applied egg-rr 47.9%

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

      1. unpow1 47.9%

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

      2. associate-/l* 44.8%

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

      3. *-commutative 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in d around -inf 61.7%

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

       1. associate-*r* 61.7%

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

       2. mul-1-neg 61.7%

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

   12. Simplified 61.7%

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

       1. pow1 61.7%

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

       2. associate-*l* 62.0%

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

       3. sqrt-div 62.7%

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

       4. metadata-eval 62.7%

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

       5. associate-/r/ 71.7%

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

   14. Applied egg-rr 71.7%

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

       1. unpow1 71.7%

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

       2. *-commutative 71.7%

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

       3. associate-*l/ 71.7%

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

   16. Simplified 71.7%

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

   if -6.49999999999999963e-296 < l < 1.54999999999999987e120

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. metadata-eval 74.4%

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

      3. unpow1/2 74.4%

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

      4. metadata-eval 74.4%

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

      5. unpow1/2 74.4%

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

      6. associate-*l* 74.4%

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

      7. metadata-eval 74.4%

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

      8. times-frac 74.4%

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

   3. Simplified 74.4%

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

   if 1.54999999999999987e120 < l

   1. Initial program 51.9%

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

      1. metadata-eval 51.9%

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

      2. unpow1/2 51.9%

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

      3. metadata-eval 51.9%

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

      4. unpow1/2 51.9%

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

      5. *-commutative 51.9%

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

      6. associate-*l* 51.9%

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

      7. times-frac 52.2%

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

      8. metadata-eval 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in d around inf 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. *-commutative 67.2%

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

   6. Applied egg-rr 67.2%

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

      1. *-lft-identity 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-/r* 67.2%

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

   8. Simplified 67.2%

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

      1. sqrt-div 76.3%

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

   10. Applied egg-rr 76.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 4: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+125}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.5e-296)
   (*
    d
    (*
     (/ 1.0 (sqrt (* l h)))
     (+ (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* h 0.5)) l) -1.0)))
   (if (<= l 1.25e+125)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (pow (* (/ M d) (/ D 2.0)) 2.0) (* 0.5 (/ h l)))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.5e-296) {
		tmp = d * ((1.0 / sqrt((l * h))) * (((pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1.25e+125) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (pow(((M / d) * (D / 2.0)), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.5d-296)) then
        tmp = d * ((1.0d0 / sqrt((l * h))) * (((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0)))
    else if (l <= 1.25d+125) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((m / d) * (d_1 / 2.0d0)) ** 2.0d0) * (0.5d0 * (h / l))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.5e-296) {
		tmp = d * ((1.0 / Math.sqrt((l * h))) * (((Math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1.25e+125) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (Math.pow(((M / d) * (D / 2.0)), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.5e-296:
		tmp = d * ((1.0 / math.sqrt((l * h))) * (((math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0))
	elif l <= 1.25e+125:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (math.pow(((M / d) * (D / 2.0)), 2.0) * (0.5 * (h / l))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.5e-296)
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(Float64(l * h))) * Float64(Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(h * 0.5)) / l) + -1.0)));
	elseif (l <= 1.25e+125)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64((Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.5e-296)
		tmp = d * ((1.0 / sqrt((l * h))) * (((((M * (0.5 * (D / d))) ^ 2.0) * (h * 0.5)) / l) + -1.0));
	elseif (l <= 1.25e+125)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((M / d) * (D / 2.0)) ^ 2.0) * (0.5 * (h / l))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.5e-296], N[(d * N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+125], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.49999999999999963e-296

   1. Initial program 59.4%

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

      1. metadata-eval 59.4%

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

      2. unpow1/2 59.4%

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

      3. metadata-eval 59.4%

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

      4. unpow1/2 59.4%

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

      5. *-commutative 59.4%

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

      6. associate-*l* 59.4%

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

      7. times-frac 60.0%

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

      8. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

      1. associate-*r* 60.0%

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

      2. frac-times 59.4%

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

      3. *-commutative 59.4%

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

      4. metadata-eval 59.4%

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

      5. associate-*r/ 62.0%

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

      6. metadata-eval 62.0%

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

      7. *-commutative 62.0%

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

      8. frac-times 62.5%

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

      9. div-inv 62.5%

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

      10. metadata-eval 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. pow1 62.5%

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

      2. sqrt-unprod 47.9%

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

      3. associate-*l* 47.9%

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

      4. associate-*l* 47.9%

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

   7. Applied egg-rr 47.9%

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

      1. unpow1 47.9%

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

      2. associate-/l* 44.8%

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

      3. *-commutative 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in d around -inf 61.7%

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

       1. associate-*r* 61.7%

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

       2. mul-1-neg 61.7%

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

   12. Simplified 61.7%

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

       1. pow1 61.7%

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

       2. associate-*l* 62.0%

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

       3. sqrt-div 62.7%

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

       4. metadata-eval 62.7%

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

       5. associate-/r/ 71.7%

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

   14. Applied egg-rr 71.7%

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

       1. unpow1 71.7%

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

       2. *-commutative 71.7%

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

       3. associate-*l/ 71.7%

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

   16. Simplified 71.7%

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

   if -6.49999999999999963e-296 < l < 1.24999999999999991e125

   1. Initial program 74.4%

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

      1. metadata-eval 74.4%

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

      2. unpow1/2 74.4%

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

      3. metadata-eval 74.4%

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

      4. unpow1/2 74.4%

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

      5. *-commutative 74.4%

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

      6. associate-*l* 74.4%

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

      7. times-frac 74.4%

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

      8. metadata-eval 74.4%

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

   3. Simplified 74.4%

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

      1. frac-times 74.4%

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

      2. associate-/l* 74.4%

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

      3. *-commutative 74.4%

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

   5. Applied egg-rr 74.4%

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

      1. associate-/l* 74.4%

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

      2. times-frac 72.3%

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

   7. Simplified 72.3%

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

   if 1.24999999999999991e125 < l

   1. Initial program 51.9%

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

      1. metadata-eval 51.9%

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

      2. unpow1/2 51.9%

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

      3. metadata-eval 51.9%

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

      4. unpow1/2 51.9%

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

      5. *-commutative 51.9%

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

      6. associate-*l* 51.9%

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

      7. times-frac 52.2%

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

      8. metadata-eval 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in d around inf 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. *-commutative 67.2%

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

   6. Applied egg-rr 67.2%

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

      1. *-lft-identity 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-/r* 67.2%

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

   8. Simplified 67.2%

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

      1. sqrt-div 76.3%

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

   10. Applied egg-rr 76.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+125}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 5: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+126}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.5e-296)
   (*
    d
    (*
     (/ 1.0 (sqrt (* l h)))
     (+ (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* h 0.5)) l) -1.0)))
   (if (<= l 4.1e+126)
     (*
      (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l))
      (* (sqrt (/ d l)) (sqrt (/ d h))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.5e-296) {
		tmp = d * ((1.0 / sqrt((l * h))) * (((pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 4.1e+126) {
		tmp = (1.0 - ((h * (0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.5d-296)) then
        tmp = d * ((1.0d0 / sqrt((l * h))) * (((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0)))
    else if (l <= 4.1d+126) then
        tmp = (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0))) / l)) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.5e-296) {
		tmp = d * ((1.0 / Math.sqrt((l * h))) * (((Math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 4.1e+126) {
		tmp = (1.0 - ((h * (0.5 * Math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.5e-296:
		tmp = d * ((1.0 / math.sqrt((l * h))) * (((math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0))
	elif l <= 4.1e+126:
		tmp = (1.0 - ((h * (0.5 * math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.5e-296)
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(Float64(l * h))) * Float64(Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(h * 0.5)) / l) + -1.0)));
	elseif (l <= 4.1e+126)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l)) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.5e-296)
		tmp = d * ((1.0 / sqrt((l * h))) * (((((M * (0.5 * (D / d))) ^ 2.0) * (h * 0.5)) / l) + -1.0));
	elseif (l <= 4.1e+126)
		tmp = (1.0 - ((h * (0.5 * (((D / d) * (M * 0.5)) ^ 2.0))) / l)) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.5e-296], N[(d * N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.1e+126], N[(N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.49999999999999963e-296

   1. Initial program 59.4%

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

      1. metadata-eval 59.4%

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

      2. unpow1/2 59.4%

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

      3. metadata-eval 59.4%

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

      4. unpow1/2 59.4%

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

      5. *-commutative 59.4%

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

      6. associate-*l* 59.4%

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

      7. times-frac 60.0%

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

      8. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

      1. associate-*r* 60.0%

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

      2. frac-times 59.4%

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

      3. *-commutative 59.4%

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

      4. metadata-eval 59.4%

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

      5. associate-*r/ 62.0%

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

      6. metadata-eval 62.0%

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

      7. *-commutative 62.0%

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

      8. frac-times 62.5%

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

      9. div-inv 62.5%

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

      10. metadata-eval 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. pow1 62.5%

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

      2. sqrt-unprod 47.9%

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

      3. associate-*l* 47.9%

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

      4. associate-*l* 47.9%

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

   7. Applied egg-rr 47.9%

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

      1. unpow1 47.9%

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

      2. associate-/l* 44.8%

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

      3. *-commutative 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in d around -inf 61.7%

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

       1. associate-*r* 61.7%

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

       2. mul-1-neg 61.7%

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

   12. Simplified 61.7%

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

       1. pow1 61.7%

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

       2. associate-*l* 62.0%

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

       3. sqrt-div 62.7%

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

       4. metadata-eval 62.7%

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

       5. associate-/r/ 71.7%

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

   14. Applied egg-rr 71.7%

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

       1. unpow1 71.7%

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

       2. *-commutative 71.7%

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

       3. associate-*l/ 71.7%

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

   16. Simplified 71.7%

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

   if -6.49999999999999963e-296 < l < 4.1000000000000001e126

   1. Initial program 73.6%

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

      1. metadata-eval 73.6%

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

      2. unpow1/2 73.6%

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

      3. metadata-eval 73.6%

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

      4. unpow1/2 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-*l* 73.6%

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

      7. times-frac 73.6%

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

      8. metadata-eval 73.6%

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

   3. Simplified 73.6%

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

      1. associate-*r* 73.6%

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

      2. frac-times 73.6%

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

      3. *-commutative 73.6%

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

      4. metadata-eval 73.6%

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

      5. associate-*r/ 77.8%

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

      6. metadata-eval 77.8%

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

      7. *-commutative 77.8%

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

      8. frac-times 77.8%

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

      9. div-inv 77.8%

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

      10. metadata-eval 77.8%

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

   5. Applied egg-rr 77.8%

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

   if 4.1000000000000001e126 < l

   1. Initial program 53.2%

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

      1. metadata-eval 53.2%

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

      2. unpow1/2 53.2%

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

      3. metadata-eval 53.2%

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

      4. unpow1/2 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-*l* 53.2%

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

      7. times-frac 53.4%

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

      8. metadata-eval 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in d around inf 68.8%

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

      1. *-un-lft-identity 68.8%

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

      2. *-commutative 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. *-lft-identity 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-/r* 68.8%

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

   8. Simplified 68.8%

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

      1. sqrt-div 78.1%

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+126}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 6: 70.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.95 \cdot 10^{-295}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-122}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot \left(h \cdot D\right)}{d} \cdot \frac{M \cdot M}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.95e-295)
   (*
    d
    (*
     (/ 1.0 (sqrt (* l h)))
     (+ (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* h 0.5)) l) -1.0)))
   (if (<= l 1e-122)
     (*
      (sqrt (* d (/ (/ d h) l)))
      (- 1.0 (* 0.5 (* h (/ (pow (* D (/ (* M 0.5) d)) 2.0) l)))))
     (if (<= l 2.8e+53)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (- 1.0 (* 0.125 (* (/ (* D (* h D)) d) (/ (* M M) (* l d))))))
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.95e-295) {
		tmp = d * ((1.0 / sqrt((l * h))) * (((pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1e-122) {
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else if (l <= 2.8e+53) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.125 * (((D * (h * D)) / d) * ((M * M) / (l * d)))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.95d-295)) then
        tmp = d * ((1.0d0 / sqrt((l * h))) * (((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0)))
    else if (l <= 1d-122) then
        tmp = sqrt((d * ((d / h) / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * ((m * 0.5d0) / d)) ** 2.0d0) / l))))
    else if (l <= 2.8d+53) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.125d0 * (((d_1 * (h * d_1)) / d) * ((m * m) / (l * d)))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.95e-295) {
		tmp = d * ((1.0 / Math.sqrt((l * h))) * (((Math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 1e-122) {
		tmp = Math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (Math.pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else if (l <= 2.8e+53) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.125 * (((D * (h * D)) / d) * ((M * M) / (l * d)))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.95e-295:
		tmp = d * ((1.0 / math.sqrt((l * h))) * (((math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0))
	elif l <= 1e-122:
		tmp = math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (math.pow((D * ((M * 0.5) / d)), 2.0) / l))))
	elif l <= 2.8e+53:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.125 * (((D * (h * D)) / d) * ((M * M) / (l * d)))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.95e-295)
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(Float64(l * h))) * Float64(Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(h * 0.5)) / l) + -1.0)));
	elseif (l <= 1e-122)
		tmp = Float64(sqrt(Float64(d * Float64(Float64(d / h) / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) / l)))));
	elseif (l <= 2.8e+53)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(D * Float64(h * D)) / d) * Float64(Float64(M * M) / Float64(l * d))))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.95e-295)
		tmp = d * ((1.0 / sqrt((l * h))) * (((((M * (0.5 * (D / d))) ^ 2.0) * (h * 0.5)) / l) + -1.0));
	elseif (l <= 1e-122)
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (((D * ((M * 0.5) / d)) ^ 2.0) / l))));
	elseif (l <= 2.8e+53)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.125 * (((D * (h * D)) / d) * ((M * M) / (l * d)))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.95e-295], N[(d * N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e-122], N[(N[Sqrt[N[(d * N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e+53], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.95000000000000006e-295

   1. Initial program 59.9%

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

      1. metadata-eval 59.9%

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

      2. unpow1/2 59.9%

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

      3. metadata-eval 59.9%

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

      4. unpow1/2 59.9%

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

      5. *-commutative 59.9%

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

      6. associate-*l* 59.9%

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

      7. times-frac 60.5%

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

      8. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

      1. associate-*r* 60.5%

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

      2. frac-times 59.9%

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

      3. *-commutative 59.9%

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

      4. metadata-eval 59.9%

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

      5. associate-*r/ 61.7%

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

      6. metadata-eval 61.7%

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

      7. *-commutative 61.7%

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

      8. frac-times 62.2%

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

      9. div-inv 62.2%

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

      10. metadata-eval 62.2%

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

   5. Applied egg-rr 62.2%

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

      1. pow1 62.2%

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

      2. sqrt-unprod 47.4%

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

      3. associate-*l* 47.4%

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

      4. associate-*l* 47.4%

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

   7. Applied egg-rr 47.4%

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

      1. unpow1 47.4%

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

      2. associate-/l* 45.2%

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

      3. *-commutative 45.2%

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

   9. Simplified 45.2%

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

   10. Taylor expanded in d around -inf 62.2%

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

       1. associate-*r* 62.2%

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

       2. mul-1-neg 62.2%

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

   12. Simplified 62.2%

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

       1. pow1 62.2%

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

       2. associate-*l* 62.5%

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

       3. sqrt-div 63.2%

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

       4. metadata-eval 63.2%

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

       5. associate-/r/ 71.4%

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

   14. Applied egg-rr 71.4%

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

       1. unpow1 71.4%

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

       2. *-commutative 71.4%

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

       3. associate-*l/ 71.5%

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

   16. Simplified 71.5%

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

   if -2.95000000000000006e-295 < l < 1.00000000000000006e-122

   1. Initial program 74.9%

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

      1. metadata-eval 74.9%

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

      2. unpow1/2 74.9%

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

      3. metadata-eval 74.9%

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

      4. unpow1/2 74.9%

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

      5. *-commutative 74.9%

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

      6. associate-*l* 74.9%

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

      7. times-frac 74.9%

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

      8. metadata-eval 74.9%

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

   3. Simplified 74.9%

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

      1. associate-*r* 74.9%

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

      2. frac-times 74.9%

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

      3. *-commutative 74.9%

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

      4. metadata-eval 74.9%

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

      5. associate-*r/ 83.6%

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

      6. metadata-eval 83.6%

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

      7. *-commutative 83.6%

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

      8. frac-times 83.6%

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

      9. div-inv 83.6%

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

      10. metadata-eval 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. pow1/2 83.6%

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

      2. sqr-pow 83.5%

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

      3. pow2 83.5%

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

      4. metadata-eval 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. pow1 83.5%

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

      2. *-commutative 83.5%

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

      3. associate-*l* 83.5%

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

      4. associate-*r* 83.5%

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

      5. *-commutative 83.5%

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

      6. pow-pow 83.6%

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

      7. metadata-eval 83.6%

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

      8. pow1/2 83.6%

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

      9. sqrt-prod 77.2%

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

      10. associate-*r/ 70.5%

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

   9. Applied egg-rr 70.5%

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

       1. unpow1 70.5%

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

       2. *-commutative 70.5%

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

       3. associate-/l* 77.2%

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

       4. associate-/r/ 77.3%

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

       5. associate-*l/ 75.1%

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

       6. *-commutative 75.1%

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

       7. *-commutative 75.1%

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

       8. associate-*l* 75.1%

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

       9. associate-*r* 75.1%

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

       10. associate-*r/ 75.1%

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

       11. associate-/l* 75.1%

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

       12. associate-/r/ 73.0%

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

   11. Simplified 73.0%

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

   if 1.00000000000000006e-122 < l < 2.8e53

   1. Initial program 75.5%

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

      1. metadata-eval 75.5%

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

      2. unpow1/2 75.5%

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

      3. metadata-eval 75.5%

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

      4. unpow1/2 75.5%

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

      5. *-commutative 75.5%

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

      6. associate-*l* 75.5%

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

      7. times-frac 75.5%

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

      8. metadata-eval 75.5%

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

   3. Simplified 75.5%

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

      1. frac-times 75.5%

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

      2. associate-/l* 75.5%

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

      3. *-commutative 75.5%

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

   5. Applied egg-rr 75.5%

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

      1. associate-/l* 75.5%

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

      2. times-frac 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in M around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*r* 54.9%

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

      4. unpow2 54.9%

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

      5. associate-*l* 55.2%

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

      6. times-frac 63.2%

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

      7. unpow2 63.2%

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

      8. associate-*l* 66.1%

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

      9. unpow2 66.1%

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

   10. Simplified 66.1%

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

   if 2.8e53 < l

   1. Initial program 54.8%

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

      1. metadata-eval 54.8%

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

      2. unpow1/2 54.8%

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

      3. metadata-eval 54.8%

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

      4. unpow1/2 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*l* 54.8%

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

      7. times-frac 55.0%

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

      8. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in d around inf 63.1%

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

      1. *-un-lft-identity 63.1%

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

      2. *-commutative 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. *-lft-identity 63.1%

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

      2. *-commutative 63.1%

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

      3. associate-/r* 63.0%

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

   8. Simplified 63.0%

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

      1. sqrt-div 72.9%

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

   10. Applied egg-rr 72.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.95 \cdot 10^{-295}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-122}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot \left(h \cdot D\right)}{d} \cdot \frac{M \cdot M}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 7: 67.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-215}:\\ \;\;\;\;\frac{d \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(h \cdot 0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e-215)
   (/
    (* d (- -1.0 (/ -0.5 (/ (/ l h) (pow (/ (* D (* M 0.5)) d) 2.0)))))
    (sqrt (* l h)))
   (if (<= l 1.35e+53)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (- 1.0 (* (* h 0.5) (/ (pow (* M (* 0.5 (/ D d))) 2.0) l))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-215) {
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / pow(((D * (M * 0.5)) / d), 2.0))))) / sqrt((l * h));
	} else if (l <= 1.35e+53) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h * 0.5) * (pow((M * (0.5 * (D / d))), 2.0) / l)));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-215)) then
        tmp = (d * ((-1.0d0) - ((-0.5d0) / ((l / h) / (((d_1 * (m * 0.5d0)) / d) ** 2.0d0))))) / sqrt((l * h))
    else if (l <= 1.35d+53) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((h * 0.5d0) * (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / l)))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-215) {
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / Math.pow(((D * (M * 0.5)) / d), 2.0))))) / Math.sqrt((l * h));
	} else if (l <= 1.35e+53) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - ((h * 0.5) * (Math.pow((M * (0.5 * (D / d))), 2.0) / l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e-215:
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / math.pow(((D * (M * 0.5)) / d), 2.0))))) / math.sqrt((l * h))
	elif l <= 1.35e+53:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - ((h * 0.5) * (math.pow((M * (0.5 * (D / d))), 2.0) / l)))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e-215)
		tmp = Float64(Float64(d * Float64(-1.0 - Float64(-0.5 / Float64(Float64(l / h) / (Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0))))) / sqrt(Float64(l * h)));
	elseif (l <= 1.35e+53)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(h * 0.5) * Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) / l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e-215)
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / (((D * (M * 0.5)) / d) ^ 2.0))))) / sqrt((l * h));
	elseif (l <= 1.35e+53)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((h * 0.5) * (((M * (0.5 * (D / d))) ^ 2.0) / l)));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e-215], N[(N[(d * N[(-1.0 - N[(-0.5 / N[(N[(l / h), $MachinePrecision] / N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e+53], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h * 0.5), $MachinePrecision] * N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.00000000000000004e-215

   1. Initial program 58.9%

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

      1. metadata-eval 58.9%

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

      2. unpow1/2 58.9%

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

      3. metadata-eval 58.9%

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

      4. unpow1/2 58.9%

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

      5. *-commutative 58.9%

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

      6. associate-*l* 58.9%

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

      7. times-frac 59.0%

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

      8. metadata-eval 59.0%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.0%

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

      2. frac-times 58.9%

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

      3. *-commutative 58.9%

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

      4. metadata-eval 58.9%

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

      5. associate-*r/ 58.9%

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

      6. metadata-eval 58.9%

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

      7. *-commutative 58.9%

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

      8. frac-times 59.0%

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

      9. div-inv 59.0%

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

      10. metadata-eval 59.0%

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

   5. Applied egg-rr 59.0%

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

      1. pow1 59.0%

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

      2. sqrt-unprod 42.4%

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

      3. associate-*l* 42.4%

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

      4. associate-*l* 42.4%

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

   7. Applied egg-rr 42.4%

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

      1. unpow1 42.4%

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

      2. associate-/l* 41.8%

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

      3. *-commutative 41.8%

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

   9. Simplified 41.8%

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

   10. Taylor expanded in d around -inf 61.0%

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

       1. associate-*r* 61.0%

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

       2. mul-1-neg 61.0%

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

   12. Simplified 61.0%

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

       1. pow1 61.0%

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

       2. associate-*l* 62.0%

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

       3. sqrt-div 62.9%

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

       4. metadata-eval 62.9%

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

       5. associate-/r/ 68.8%

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

   14. Applied egg-rr 68.8%

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

       1. unpow1 68.8%

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

       2. associate-*l/ 68.7%

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

       3. *-lft-identity 68.7%

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

       4. associate-*r/ 68.7%

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

   16. Simplified 61.9%

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

   if -1.00000000000000004e-215 < l < 1.3500000000000001e53

   1. Initial program 73.2%

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

      1. metadata-eval 73.2%

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

      2. unpow1/2 73.2%

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

      3. metadata-eval 73.2%

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

      4. unpow1/2 73.2%

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

      5. *-commutative 73.2%

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

      6. associate-*l* 73.2%

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

      7. times-frac 73.9%

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

      8. metadata-eval 73.9%

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

   3. Simplified 73.9%

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

      1. associate-*r* 73.9%

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

      2. frac-times 73.2%

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

      3. *-commutative 73.2%

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

      4. metadata-eval 73.2%

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

      5. associate-*r/ 79.3%

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

      6. metadata-eval 79.3%

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

      7. *-commutative 79.3%

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

      8. frac-times 80.0%

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

      9. div-inv 80.0%

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

      10. metadata-eval 80.0%

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

   5. Applied egg-rr 80.0%

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

      1. pow1 80.0%

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

      2. sqrt-unprod 67.8%

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

      3. associate-*l* 67.8%

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

      4. associate-*l* 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

      2. associate-/l* 61.8%

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

      3. *-commutative 61.8%

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

   9. Simplified 61.8%

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

       1. associate-/r/ 66.8%

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

   11. Applied egg-rr 66.8%

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

   if 1.3500000000000001e53 < l

   1. Initial program 54.8%

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

      1. metadata-eval 54.8%

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

      2. unpow1/2 54.8%

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

      3. metadata-eval 54.8%

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

      4. unpow1/2 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*l* 54.8%

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

      7. times-frac 55.0%

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

      8. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in d around inf 63.1%

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

      1. *-un-lft-identity 63.1%

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

      2. *-commutative 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. *-lft-identity 63.1%

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

      2. *-commutative 63.1%

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

      3. associate-/r* 63.0%

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

   8. Simplified 63.0%

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

      1. sqrt-div 72.9%

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

   10. Applied egg-rr 72.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-215}:\\ \;\;\;\;\frac{d \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \left(h \cdot 0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 8: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.07 \cdot 10^{-215}:\\ \;\;\;\;\frac{d \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.07e-215)
   (/
    (* d (- -1.0 (/ -0.5 (/ (/ l h) (pow (/ (* D (* M 0.5)) d) 2.0)))))
    (sqrt (* l h)))
   (if (<= l 1.4e+52)
     (*
      (sqrt (* d (/ (/ d h) l)))
      (- 1.0 (* 0.5 (* h (/ (pow (* D (/ (* M 0.5) d)) 2.0) l)))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.07e-215) {
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / pow(((D * (M * 0.5)) / d), 2.0))))) / sqrt((l * h));
	} else if (l <= 1.4e+52) {
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.07d-215)) then
        tmp = (d * ((-1.0d0) - ((-0.5d0) / ((l / h) / (((d_1 * (m * 0.5d0)) / d) ** 2.0d0))))) / sqrt((l * h))
    else if (l <= 1.4d+52) then
        tmp = sqrt((d * ((d / h) / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * ((m * 0.5d0) / d)) ** 2.0d0) / l))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.07e-215) {
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / Math.pow(((D * (M * 0.5)) / d), 2.0))))) / Math.sqrt((l * h));
	} else if (l <= 1.4e+52) {
		tmp = Math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (Math.pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.07e-215:
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / math.pow(((D * (M * 0.5)) / d), 2.0))))) / math.sqrt((l * h))
	elif l <= 1.4e+52:
		tmp = math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (math.pow((D * ((M * 0.5) / d)), 2.0) / l))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.07e-215)
		tmp = Float64(Float64(d * Float64(-1.0 - Float64(-0.5 / Float64(Float64(l / h) / (Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0))))) / sqrt(Float64(l * h)));
	elseif (l <= 1.4e+52)
		tmp = Float64(sqrt(Float64(d * Float64(Float64(d / h) / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) / l)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.07e-215)
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / (((D * (M * 0.5)) / d) ^ 2.0))))) / sqrt((l * h));
	elseif (l <= 1.4e+52)
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (((D * ((M * 0.5) / d)) ^ 2.0) / l))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.07e-215], N[(N[(d * N[(-1.0 - N[(-0.5 / N[(N[(l / h), $MachinePrecision] / N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.4e+52], N[(N[Sqrt[N[(d * N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.07000000000000003e-215

   1. Initial program 58.9%

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

      1. metadata-eval 58.9%

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

      2. unpow1/2 58.9%

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

      3. metadata-eval 58.9%

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

      4. unpow1/2 58.9%

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

      5. *-commutative 58.9%

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

      6. associate-*l* 58.9%

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

      7. times-frac 59.0%

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

      8. metadata-eval 59.0%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.0%

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

      2. frac-times 58.9%

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

      3. *-commutative 58.9%

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

      4. metadata-eval 58.9%

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

      5. associate-*r/ 58.9%

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

      6. metadata-eval 58.9%

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

      7. *-commutative 58.9%

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

      8. frac-times 59.0%

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

      9. div-inv 59.0%

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

      10. metadata-eval 59.0%

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

   5. Applied egg-rr 59.0%

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

      1. pow1 59.0%

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

      2. sqrt-unprod 42.4%

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

      3. associate-*l* 42.4%

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

      4. associate-*l* 42.4%

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

   7. Applied egg-rr 42.4%

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

      1. unpow1 42.4%

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

      2. associate-/l* 41.8%

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

      3. *-commutative 41.8%

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

   9. Simplified 41.8%

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

   10. Taylor expanded in d around -inf 61.0%

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

       1. associate-*r* 61.0%

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

       2. mul-1-neg 61.0%

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

   12. Simplified 61.0%

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

       1. pow1 61.0%

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

       2. associate-*l* 62.0%

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

       3. sqrt-div 62.9%

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

       4. metadata-eval 62.9%

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

       5. associate-/r/ 68.8%

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

   14. Applied egg-rr 68.8%

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

       1. unpow1 68.8%

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

       2. associate-*l/ 68.7%

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

       3. *-lft-identity 68.7%

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

       4. associate-*r/ 68.7%

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

   16. Simplified 61.9%

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

   if -1.07000000000000003e-215 < l < 1.4e52

   1. Initial program 73.2%

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

      1. metadata-eval 73.2%

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

      2. unpow1/2 73.2%

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

      3. metadata-eval 73.2%

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

      4. unpow1/2 73.2%

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

      5. *-commutative 73.2%

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

      6. associate-*l* 73.2%

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

      7. times-frac 73.9%

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

      8. metadata-eval 73.9%

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

   3. Simplified 73.9%

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

      1. associate-*r* 73.9%

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

      2. frac-times 73.2%

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

      3. *-commutative 73.2%

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

      4. metadata-eval 73.2%

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

      5. associate-*r/ 79.3%

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

      6. metadata-eval 79.3%

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

      7. *-commutative 79.3%

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

      8. frac-times 80.0%

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

      9. div-inv 80.0%

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

      10. metadata-eval 80.0%

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

   5. Applied egg-rr 80.0%

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

      1. pow1/2 80.0%

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

      2. sqr-pow 79.7%

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

      3. pow2 79.7%

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

      4. metadata-eval 79.7%

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

   7. Applied egg-rr 79.7%

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

      1. pow1 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*l* 79.7%

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

      4. associate-*r* 79.7%

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

      5. *-commutative 79.7%

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

      6. pow-pow 80.0%

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

      7. metadata-eval 80.0%

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

      8. pow1/2 80.0%

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

      9. sqrt-prod 67.8%

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

      10. associate-*r/ 59.4%

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

   9. Applied egg-rr 59.4%

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

       1. unpow1 59.4%

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

       2. *-commutative 59.4%

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

       3. associate-/l* 67.7%

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

       4. associate-/r/ 67.8%

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

       5. associate-*l/ 66.8%

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

       6. *-commutative 66.8%

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

       7. *-commutative 66.8%

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

       8. associate-*l* 66.8%

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

       9. associate-*r* 66.8%

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

       10. associate-*r/ 66.2%

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

       11. associate-/l* 66.8%

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

       12. associate-/r/ 64.9%

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

   11. Simplified 64.9%

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

   if 1.4e52 < l

   1. Initial program 54.8%

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

      1. metadata-eval 54.8%

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

      2. unpow1/2 54.8%

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

      3. metadata-eval 54.8%

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

      4. unpow1/2 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*l* 54.8%

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

      7. times-frac 55.0%

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

      8. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in d around inf 63.1%

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

      1. *-un-lft-identity 63.1%

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

      2. *-commutative 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. *-lft-identity 63.1%

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

      2. *-commutative 63.1%

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

      3. associate-/r* 63.0%

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

   8. Simplified 63.0%

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

      1. sqrt-div 72.9%

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

   10. Applied egg-rr 72.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.07 \cdot 10^{-215}:\\ \;\;\;\;\frac{d \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 9: 66.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-215}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}} + -1\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e-215)
   (*
    (* d (pow (* l h) -0.5))
    (+ (/ (pow (* M (* 0.5 (/ D d))) 2.0) (/ l (* h 0.5))) -1.0))
   (if (<= l 9.5e+50)
     (*
      (sqrt (* d (/ (/ d h) l)))
      (- 1.0 (* 0.5 (* h (/ (pow (* D (/ (* M 0.5) d)) 2.0) l)))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-215) {
		tmp = (d * pow((l * h), -0.5)) * ((pow((M * (0.5 * (D / d))), 2.0) / (l / (h * 0.5))) + -1.0);
	} else if (l <= 9.5e+50) {
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-215)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / (l / (h * 0.5d0))) + (-1.0d0))
    else if (l <= 9.5d+50) then
        tmp = sqrt((d * ((d / h) / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * ((m * 0.5d0) / d)) ** 2.0d0) / l))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-215) {
		tmp = (d * Math.pow((l * h), -0.5)) * ((Math.pow((M * (0.5 * (D / d))), 2.0) / (l / (h * 0.5))) + -1.0);
	} else if (l <= 9.5e+50) {
		tmp = Math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (Math.pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e-215:
		tmp = (d * math.pow((l * h), -0.5)) * ((math.pow((M * (0.5 * (D / d))), 2.0) / (l / (h * 0.5))) + -1.0)
	elif l <= 9.5e+50:
		tmp = math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (math.pow((D * ((M * 0.5) / d)), 2.0) / l))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e-215)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) / Float64(l / Float64(h * 0.5))) + -1.0));
	elseif (l <= 9.5e+50)
		tmp = Float64(sqrt(Float64(d * Float64(Float64(d / h) / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) / l)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e-215)
		tmp = (d * ((l * h) ^ -0.5)) * ((((M * (0.5 * (D / d))) ^ 2.0) / (l / (h * 0.5))) + -1.0);
	elseif (l <= 9.5e+50)
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (((D * ((M * 0.5) / d)) ^ 2.0) / l))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e-215], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.5e+50], N[(N[Sqrt[N[(d * N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.00000000000000004e-215

   1. Initial program 58.9%

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

      1. metadata-eval 58.9%

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

      2. unpow1/2 58.9%

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

      3. metadata-eval 58.9%

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

      4. unpow1/2 58.9%

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

      5. *-commutative 58.9%

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

      6. associate-*l* 58.9%

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

      7. times-frac 59.0%

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

      8. metadata-eval 59.0%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.0%

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

      2. frac-times 58.9%

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

      3. *-commutative 58.9%

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

      4. metadata-eval 58.9%

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

      5. associate-*r/ 58.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 58.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 58.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 59.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 59.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 59.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 59.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 59.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 42.4%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 42.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 42.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 42.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 41.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 41.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 41.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 61.0%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg 61.0%

          \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. associate-/l/ 61.5%

          \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       3. distribute-rgt-neg-in 61.5%

          \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       4. associate-/l/ 61.0%

          \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       5. unpow-1 61.0%

          \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       6. metadata-eval 61.0%

          \[\leadsto \left(d \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       7. pow-sqr 61.0%

          \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       8. rem-sqrt-square 61.9%

          \[\leadsto \left(d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|}\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       9. rem-square-sqrt 61.8%

          \[\leadsto \left(d \cdot \left(-\left|\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right|\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       10. fabs-sqr 61.8%

           \[\leadsto \left(d \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       11. rem-square-sqrt 61.9%

           \[\leadsto \left(d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right)\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 61.9%

       \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   if -1.00000000000000004e-215 < l < 9.4999999999999993e50

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 80.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 80.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1/2 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      2. sqr-pow 79.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      3. pow2 79.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      4. metadata-eval 79.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   7. Applied egg-rr 79.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. pow1 79.7%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 79.7%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}}^{1} \]

      3. associate-*l* 79.7%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}^{1} \]

      4. associate-*r* 79.7%

         \[\leadsto {\left(\left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}^{1} \]

      5. *-commutative 79.7%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(h \cdot 0.5\right)}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}^{1} \]

      6. pow-pow 80.0%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(0.25 \cdot 2\right)}}\right)\right)}^{1} \]

      7. metadata-eval 80.0%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)\right)}^{1} \]

      8. pow1/2 80.0%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)\right)}^{1} \]

      9. sqrt-prod 67.8%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      10. associate-*r/ 59.4%

          \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}}\right)}^{1} \]

   9. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \sqrt{\frac{\frac{d}{h} \cdot d}{\ell}}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 59.4%

          \[\leadsto \color{blue}{\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \sqrt{\frac{\frac{d}{h} \cdot d}{\ell}}} \]

       2. *-commutative 59.4%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)} \]

       3. associate-/l* 67.7%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \]

       4. associate-/r/ 67.8%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\ell} \cdot d}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \]

       5. associate-*l/ 66.8%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)}\right) \]

       6. *-commutative 66.8%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\left(h \cdot 0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right) \]

       7. *-commutative 66.8%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\left(0.5 \cdot h\right)} \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \]

       8. associate-*l* 66.8%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]

       9. associate-*r* 66.8%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2}}{\ell}\right)\right) \]

       10. associate-*r/ 66.2%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2}}{\ell}\right)\right) \]

       11. associate-/l* 66.8%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}}^{2}}{\ell}\right)\right) \]

       12. associate-/r/ 64.9%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}}^{2}}{\ell}\right)\right) \]

   11. Simplified 64.9%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}^{2}}{\ell}\right)\right)} \]

   if 9.4999999999999993e50 < l

   1. Initial program 54.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 54.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 54.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 55.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 55.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 55.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 63.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 63.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 63.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 63.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 63.1%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 63.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. sqrt-div 72.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]

   10. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-215}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}} + -1\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 10: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{-295}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.4e-295)
   (*
    d
    (*
     (/ 1.0 (sqrt (* l h)))
     (+ (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* h 0.5)) l) -1.0)))
   (if (<= l 3.8e+51)
     (*
      (sqrt (* d (/ (/ d h) l)))
      (- 1.0 (* 0.5 (* h (/ (pow (* D (/ (* M 0.5) d)) 2.0) l)))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.4e-295) {
		tmp = d * ((1.0 / sqrt((l * h))) * (((pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 3.8e+51) {
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.4d-295)) then
        tmp = d * ((1.0d0 / sqrt((l * h))) * (((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0)))
    else if (l <= 3.8d+51) then
        tmp = sqrt((d * ((d / h) / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * ((m * 0.5d0) / d)) ** 2.0d0) / l))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.4e-295) {
		tmp = d * ((1.0 / Math.sqrt((l * h))) * (((Math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0));
	} else if (l <= 3.8e+51) {
		tmp = Math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (Math.pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.4e-295:
		tmp = d * ((1.0 / math.sqrt((l * h))) * (((math.pow((M * (0.5 * (D / d))), 2.0) * (h * 0.5)) / l) + -1.0))
	elif l <= 3.8e+51:
		tmp = math.sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (math.pow((D * ((M * 0.5) / d)), 2.0) / l))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.4e-295)
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(Float64(l * h))) * Float64(Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(h * 0.5)) / l) + -1.0)));
	elseif (l <= 3.8e+51)
		tmp = Float64(sqrt(Float64(d * Float64(Float64(d / h) / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) / l)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.4e-295)
		tmp = d * ((1.0 / sqrt((l * h))) * (((((M * (0.5 * (D / d))) ^ 2.0) * (h * 0.5)) / l) + -1.0));
	elseif (l <= 3.8e+51)
		tmp = sqrt((d * ((d / h) / l))) * (1.0 - (0.5 * (h * (((D * ((M * 0.5) / d)) ^ 2.0) / l))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.4e-295], N[(d * N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e+51], N[(N[Sqrt[N[(d * N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.4e-295

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 59.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 59.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 60.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 62.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 62.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 47.4%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 47.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 47.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 47.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 47.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 45.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 45.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 45.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 62.2%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 62.2%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 62.2%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 62.2%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   13. Step-by-step derivation

       1. pow1 62.2%

          \[\leadsto \color{blue}{{\left(\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)}^{1}} \]

       2. associate-*l* 62.5%

          \[\leadsto {\color{blue}{\left(\left(-d\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)\right)}}^{1} \]

       3. sqrt-div 63.2%

          \[\leadsto {\left(\left(-d\right) \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)\right)}^{1} \]

       4. metadata-eval 63.2%

          \[\leadsto {\left(\left(-d\right) \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)\right)}^{1} \]

       5. associate-/r/ 71.4%

          \[\leadsto {\left(\left(-d\right) \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)}\right)\right)\right)}^{1} \]

   14. Applied egg-rr 71.4%

       \[\leadsto \color{blue}{{\left(\left(-d\right) \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)\right)\right)}^{1}} \]
   15. Step-by-step derivation

       1. unpow1 71.4%

          \[\leadsto \color{blue}{\left(-d\right) \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)\right)} \]

       2. *-commutative 71.4%

          \[\leadsto \left(-d\right) \cdot \left(\frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)\right) \]

       3. associate-*l/ 71.5%

          \[\leadsto \left(-d\right) \cdot \left(\frac{1}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}}\right)\right) \]

   16. Simplified 71.5%

       \[\leadsto \color{blue}{\left(-d\right) \cdot \left(\frac{1}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\right)} \]

   if -6.4e-295 < l < 3.7999999999999997e51

   1. Initial program 75.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 75.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 75.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 80.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 80.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1/2 80.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      2. sqr-pow 79.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      3. pow2 79.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      4. metadata-eval 79.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   7. Applied egg-rr 79.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
   8. Step-by-step derivation

      1. pow1 79.8%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 79.8%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}}^{1} \]

      3. associate-*l* 79.8%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}^{1} \]

      4. associate-*r* 79.8%

         \[\leadsto {\left(\left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}^{1} \]

      5. *-commutative 79.8%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(h \cdot 0.5\right)}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)\right)}^{1} \]

      6. pow-pow 80.0%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(0.25 \cdot 2\right)}}\right)\right)}^{1} \]

      7. metadata-eval 80.0%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)\right)}^{1} \]

      8. pow1/2 80.0%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)\right)}^{1} \]

      9. sqrt-prod 66.1%

         \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      10. associate-*r/ 60.8%

          \[\leadsto {\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}}\right)}^{1} \]

   9. Applied egg-rr 60.8%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \sqrt{\frac{\frac{d}{h} \cdot d}{\ell}}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 60.8%

          \[\leadsto \color{blue}{\left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \sqrt{\frac{\frac{d}{h} \cdot d}{\ell}}} \]

       2. *-commutative 60.8%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)} \]

       3. associate-/l* 66.1%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \]

       4. associate-/r/ 66.1%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\ell} \cdot d}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \]

       5. associate-*l/ 65.0%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)}\right) \]

       6. *-commutative 65.0%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\left(h \cdot 0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right) \]

       7. *-commutative 65.0%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\left(0.5 \cdot h\right)} \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \]

       8. associate-*l* 65.0%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - \color{blue}{0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]

       9. associate-*r* 65.0%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2}}{\ell}\right)\right) \]

       10. associate-*r/ 65.0%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2}}{\ell}\right)\right) \]

       11. associate-/l* 65.0%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}}^{2}}{\ell}\right)\right) \]

       12. associate-/r/ 62.6%

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}}^{2}}{\ell}\right)\right) \]

   11. Simplified 62.6%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\ell} \cdot d} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}^{2}}{\ell}\right)\right)} \]

   if 3.7999999999999997e51 < l

   1. Initial program 54.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 54.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 54.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 55.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 55.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 55.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 63.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 63.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 63.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 63.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 63.1%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 63.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. sqrt-div 72.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]

   10. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{-295}:\\ \;\;\;\;d \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 11: 54.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ t_2 := 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\\ \mathbf{if}\;d \leq -6 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{+93}:\\ \;\;\;\;t_0 \cdot \left(1 - t_2\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t_2 + -1\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}}\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+37}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h))))
        (t_1 (* d (- (pow (* l h) -0.5))))
        (t_2 (* 0.125 (* (/ (* D D) l) (/ (* h (* M M)) (* d d))))))
   (if (<= d -6e+153)
     t_1
     (if (<= d -2.5e+93)
       (* t_0 (- 1.0 t_2))
       (if (<= d -2e-5)
         t_1
         (if (<= d -5e-310)
           (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_2 -1.0))
           (if (<= d 1.05e-125)
             (* (sqrt (/ h (pow l 3.0))) (* -0.125 (/ D (/ (/ (/ d M) M) D))))
             (if (<= d 4.6e+37)
               (*
                t_0
                (- 1.0 (* 0.125 (/ (* D (* D (* M M))) (/ (* l (* d d)) h)))))
               (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double t_1 = d * -pow((l * h), -0.5);
	double t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	double tmp;
	if (d <= -6e+153) {
		tmp = t_1;
	} else if (d <= -2.5e+93) {
		tmp = t_0 * (1.0 - t_2);
	} else if (d <= -2e-5) {
		tmp = t_1;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	} else if (d <= 1.05e-125) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D / (((d / M) / M) / D)));
	} else if (d <= 4.6e+37) {
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    t_1 = d * -((l * h) ** (-0.5d0))
    t_2 = 0.125d0 * (((d_1 * d_1) / l) * ((h * (m * m)) / (d * d)))
    if (d <= (-6d+153)) then
        tmp = t_1
    else if (d <= (-2.5d+93)) then
        tmp = t_0 * (1.0d0 - t_2)
    else if (d <= (-2d-5)) then
        tmp = t_1
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_2 + (-1.0d0))
    else if (d <= 1.05d-125) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 / (((d / m) / m) / d_1)))
    else if (d <= 4.6d+37) then
        tmp = t_0 * (1.0d0 - (0.125d0 * ((d_1 * (d_1 * (m * m))) / ((l * (d * d)) / h))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    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) * (d / h)));
	double t_1 = d * -Math.pow((l * h), -0.5);
	double t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	double tmp;
	if (d <= -6e+153) {
		tmp = t_1;
	} else if (d <= -2.5e+93) {
		tmp = t_0 * (1.0 - t_2);
	} else if (d <= -2e-5) {
		tmp = t_1;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	} else if (d <= 1.05e-125) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D / (((d / M) / M) / D)));
	} else if (d <= 4.6e+37) {
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	t_1 = d * -math.pow((l * h), -0.5)
	t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))
	tmp = 0
	if d <= -6e+153:
		tmp = t_1
	elif d <= -2.5e+93:
		tmp = t_0 * (1.0 - t_2)
	elif d <= -2e-5:
		tmp = t_1
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_2 + -1.0)
	elif d <= 1.05e-125:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D / (((d / M) / M) / D)))
	elif d <= 4.6e+37:
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_1 = Float64(d * Float64(-(Float64(l * h) ^ -0.5)))
	t_2 = Float64(0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d))))
	tmp = 0.0
	if (d <= -6e+153)
		tmp = t_1;
	elseif (d <= -2.5e+93)
		tmp = Float64(t_0 * Float64(1.0 - t_2));
	elseif (d <= -2e-5)
		tmp = t_1;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_2 + -1.0));
	elseif (d <= 1.05e-125)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D / Float64(Float64(Float64(d / M) / M) / D))));
	elseif (d <= 4.6e+37)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.125 * Float64(Float64(D * Float64(D * Float64(M * M))) / Float64(Float64(l * Float64(d * d)) / h)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	t_1 = d * -((l * h) ^ -0.5);
	t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	tmp = 0.0;
	if (d <= -6e+153)
		tmp = t_1;
	elseif (d <= -2.5e+93)
		tmp = t_0 * (1.0 - t_2);
	elseif (d <= -2e-5)
		tmp = t_1;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	elseif (d <= 1.05e-125)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D / (((d / M) / M) / D)));
	elseif (d <= 4.6e+37)
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6e+153], t$95$1, If[LessEqual[d, -2.5e+93], N[(t$95$0 * N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-5], t$95$1, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-125], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D / N[(N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+37], N[(t$95$0 * N[(1.0 - N[(0.125 * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -6.00000000000000037e153 or -2.5000000000000001e93 < d < -2.00000000000000016e-5

   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 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 64.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 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. associate-*r/ 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 73.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 54.7%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 54.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 54.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 54.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 45.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 45.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 45.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 65.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 65.8%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 65.8%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 65.8%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 65.6%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 65.6%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 65.6%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 65.6%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. unpow-1 65.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot \left(-d\right) \]

       5. metadata-eval 65.6%

          \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       6. pow-sqr 65.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \cdot \left(-d\right) \]

       7. rem-sqrt-square 67.6%

          \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       8. unpow1 67.6%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}}\right| \cdot \left(-d\right) \]

       9. sqr-pow 67.5%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(-d\right) \]

       10. fabs-sqr 67.5%

           \[\leadsto \color{blue}{\left({\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(-d\right) \]

       11. sqr-pow 67.6%

           \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}} \cdot \left(-d\right) \]

       12. unpow1 67.6%

           \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \left(-d\right) \]

       13. *-commutative 67.6%

           \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot \left(-d\right) \]

   15. Simplified 67.6%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -6.00000000000000037e153 < d < -2.5000000000000001e93

   1. Initial program 78.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 78.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 78.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 73.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 62.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 64.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 64.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 36.7%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. times-frac 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       2. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]

       3. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right) \]

       4. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right) \]

   12. Simplified 60.5%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]

   if -2.00000000000000016e-5 < d < -4.999999999999985e-310

   1. Initial program 51.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 51.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 51.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 53.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 53.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 53.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 53.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 53.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 40.3%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 40.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 40.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 40.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 40.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 40.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 40.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 40.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 59.9%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 59.9%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 59.9%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 59.9%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in M around 0 56.2%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   14. Step-by-step derivation

       1. times-frac 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       2. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]

       3. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right) \]

       4. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right) \]

   15. Simplified 54.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]

   if -4.999999999999985e-310 < d < 1.05e-125

   1. Initial program 42.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 42.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 42.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 42.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 42.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 42.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 42.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 42.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 42.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 42.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 42.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 33.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 33.3%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 33.3%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-/l* 28.2%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}}\right) \]

      4. unpow2 28.2%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}}\right) \]

      5. associate-/l* 28.8%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}}\right) \]

      6. unpow2 28.8%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}}\right) \]

      7. associate-/r* 33.6%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}}\right) \]

   8. Simplified 33.6%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}}\right)} \]

   if 1.05e-125 < d < 4.60000000000000005e37

   1. Initial program 71.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 71.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 71.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 77.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 77.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 77.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 78.0%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 78.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 78.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 78.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 78.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 71.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 71.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 71.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 60.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

       2. *-commutative 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

       3. associate-*r/ 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

       4. *-commutative 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

       5. associate-*r* 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell \cdot {d}^{2}}\right) \]

       6. associate-/l* 60.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}}\right) \]

       7. unpow2 60.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       8. associate-*l* 70.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       9. unpow2 70.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       10. unpow2 70.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \color{blue}{\left(d \cdot d\right)}}{h}}\right) \]

   12. Simplified 70.6%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}}\right) \]

   if 4.60000000000000005e37 < d

   1. Initial program 80.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 72.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 72.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 72.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 72.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 72.2%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 72.3%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 72.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. sqrt-div 82.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]

   10. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]
3. Recombined 6 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+153}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-5}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) + -1\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}}\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 12: 65.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{d \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}}\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 4.5e-308)
   (/
    (* d (- -1.0 (/ -0.5 (/ (/ l h) (pow (/ (* D (* M 0.5)) d) 2.0)))))
    (sqrt (* l h)))
   (if (<= d 7.2e-126)
     (* (sqrt (/ h (pow l 3.0))) (* -0.125 (/ D (/ (/ (/ d M) M) D))))
     (if (<= d 7.6e+37)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (- 1.0 (* 0.125 (/ (* D (* D (* M M))) (/ (* l (* d d)) h)))))
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 4.5e-308) {
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / pow(((D * (M * 0.5)) / d), 2.0))))) / sqrt((l * h));
	} else if (d <= 7.2e-126) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D / (((d / M) / M) / D)));
	} else if (d <= 7.6e+37) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(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 <= 4.5d-308) then
        tmp = (d * ((-1.0d0) - ((-0.5d0) / ((l / h) / (((d_1 * (m * 0.5d0)) / d) ** 2.0d0))))) / sqrt((l * h))
    else if (d <= 7.2d-126) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 / (((d / m) / m) / d_1)))
    else if (d <= 7.6d+37) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * ((d_1 * (d_1 * (m * m))) / ((l * (d * d)) / h))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(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 <= 4.5e-308) {
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / Math.pow(((D * (M * 0.5)) / d), 2.0))))) / Math.sqrt((l * h));
	} else if (d <= 7.2e-126) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D / (((d / M) / M) / D)));
	} else if (d <= 7.6e+37) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 4.5e-308:
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / math.pow(((D * (M * 0.5)) / d), 2.0))))) / math.sqrt((l * h))
	elif d <= 7.2e-126:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D / (((d / M) / M) / D)))
	elif d <= 7.6e+37:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 4.5e-308)
		tmp = Float64(Float64(d * Float64(-1.0 - Float64(-0.5 / Float64(Float64(l / h) / (Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0))))) / sqrt(Float64(l * h)));
	elseif (d <= 7.2e-126)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D / Float64(Float64(Float64(d / M) / M) / D))));
	elseif (d <= 7.6e+37)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D * Float64(D * Float64(M * M))) / Float64(Float64(l * Float64(d * d)) / h)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 4.5e-308)
		tmp = (d * (-1.0 - (-0.5 / ((l / h) / (((D * (M * 0.5)) / d) ^ 2.0))))) / sqrt((l * h));
	elseif (d <= 7.2e-126)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D / (((d / M) / M) / D)));
	elseif (d <= 7.6e+37)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 4.5e-308], N[(N[(d * N[(-1.0 - N[(-0.5 / N[(N[(l / h), $MachinePrecision] / N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e-126], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D / N[(N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.6e+37], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < 4.50000000000000009e-308

   1. Initial program 59.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 59.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 59.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 62.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 62.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 62.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 62.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 62.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 62.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 62.6%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 48.3%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 48.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 48.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 48.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 45.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 45.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 45.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 61.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 61.8%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 61.8%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 61.8%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   13. Step-by-step derivation

       1. pow1 61.8%

          \[\leadsto \color{blue}{{\left(\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)}^{1}} \]

       2. associate-*l* 62.1%

          \[\leadsto {\color{blue}{\left(\left(-d\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)\right)}}^{1} \]

       3. sqrt-div 62.8%

          \[\leadsto {\left(\left(-d\right) \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)\right)}^{1} \]

       4. metadata-eval 62.8%

          \[\leadsto {\left(\left(-d\right) \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)\right)\right)}^{1} \]

       5. associate-/r/ 71.6%

          \[\leadsto {\left(\left(-d\right) \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)}\right)\right)\right)}^{1} \]

   14. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{{\left(\left(-d\right) \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)\right)\right)}^{1}} \]
   15. Step-by-step derivation

       1. unpow1 71.6%

          \[\leadsto \color{blue}{\left(-d\right) \cdot \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)\right)} \]

       2. associate-*l/ 71.5%

          \[\leadsto \left(-d\right) \cdot \color{blue}{\frac{1 \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)}{\sqrt{\ell \cdot h}}} \]

       3. *-lft-identity 71.5%

          \[\leadsto \left(-d\right) \cdot \frac{\color{blue}{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)}}{\sqrt{\ell \cdot h}} \]

       4. associate-*r/ 72.1%

          \[\leadsto \color{blue}{\frac{\left(-d\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)}{\sqrt{\ell \cdot h}}} \]

   16. Simplified 62.0%

       \[\leadsto \color{blue}{\frac{\left(-d\right) \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}}}\right)}{\sqrt{h \cdot \ell}}} \]

   if 4.50000000000000009e-308 < d < 7.1999999999999999e-126

   1. Initial program 43.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 43.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 43.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 43.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 43.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 43.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 43.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 43.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 43.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 43.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 43.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 34.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 34.2%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 34.2%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-/l* 29.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}}\right) \]

      4. unpow2 29.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}}\right) \]

      5. associate-/l* 29.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}}\right) \]

      6. unpow2 29.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}}\right) \]

      7. associate-/r* 34.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}}\right) \]

   8. Simplified 34.4%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}}\right)} \]

   if 7.1999999999999999e-126 < d < 7.59999999999999979e37

   1. Initial program 71.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 71.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 71.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 71.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 77.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 77.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 77.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 78.0%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 78.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 78.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 78.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 78.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 71.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 71.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 71.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 60.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

       2. *-commutative 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

       3. associate-*r/ 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

       4. *-commutative 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

       5. associate-*r* 60.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell \cdot {d}^{2}}\right) \]

       6. associate-/l* 60.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}}\right) \]

       7. unpow2 60.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       8. associate-*l* 70.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       9. unpow2 70.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       10. unpow2 70.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \color{blue}{\left(d \cdot d\right)}}{h}}\right) \]

   12. Simplified 70.6%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}}\right) \]

   if 7.59999999999999979e37 < d

   1. Initial program 80.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 72.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 72.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 72.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 72.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 72.2%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 72.3%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 72.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. sqrt-div 82.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]

   10. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{d \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}}\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 13: 52.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ t_2 := 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\\ \mathbf{if}\;d \leq -1.05 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5.9 \cdot 10^{+93}:\\ \;\;\;\;t_0 \cdot \left(1 - t_2\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t_2 + -1\right)\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-87} \lor \neg \left(d \leq 7.8 \cdot 10^{+37}\right):\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h))))
        (t_1 (* d (- (pow (* l h) -0.5))))
        (t_2 (* 0.125 (* (/ (* D D) l) (/ (* h (* M M)) (* d d))))))
   (if (<= d -1.05e+154)
     t_1
     (if (<= d -5.9e+93)
       (* t_0 (- 1.0 t_2))
       (if (<= d -3.9e-7)
         t_1
         (if (<= d -5e-310)
           (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_2 -1.0))
           (if (or (<= d 3.7e-87) (not (<= d 7.8e+37)))
             (* d (* (pow h -0.5) (pow l -0.5)))
             (*
              t_0
              (-
               1.0
               (* 0.125 (/ (* D (* D (* M M))) (/ (* l (* d d)) h))))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double t_1 = d * -pow((l * h), -0.5);
	double t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	double tmp;
	if (d <= -1.05e+154) {
		tmp = t_1;
	} else if (d <= -5.9e+93) {
		tmp = t_0 * (1.0 - t_2);
	} else if (d <= -3.9e-7) {
		tmp = t_1;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	} else if ((d <= 3.7e-87) || !(d <= 7.8e+37)) {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	} else {
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    t_1 = d * -((l * h) ** (-0.5d0))
    t_2 = 0.125d0 * (((d_1 * d_1) / l) * ((h * (m * m)) / (d * d)))
    if (d <= (-1.05d+154)) then
        tmp = t_1
    else if (d <= (-5.9d+93)) then
        tmp = t_0 * (1.0d0 - t_2)
    else if (d <= (-3.9d-7)) then
        tmp = t_1
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_2 + (-1.0d0))
    else if ((d <= 3.7d-87) .or. (.not. (d <= 7.8d+37))) then
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    else
        tmp = t_0 * (1.0d0 - (0.125d0 * ((d_1 * (d_1 * (m * m))) / ((l * (d * d)) / h))))
    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) * (d / h)));
	double t_1 = d * -Math.pow((l * h), -0.5);
	double t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	double tmp;
	if (d <= -1.05e+154) {
		tmp = t_1;
	} else if (d <= -5.9e+93) {
		tmp = t_0 * (1.0 - t_2);
	} else if (d <= -3.9e-7) {
		tmp = t_1;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	} else if ((d <= 3.7e-87) || !(d <= 7.8e+37)) {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	} else {
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	t_1 = d * -math.pow((l * h), -0.5)
	t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))
	tmp = 0
	if d <= -1.05e+154:
		tmp = t_1
	elif d <= -5.9e+93:
		tmp = t_0 * (1.0 - t_2)
	elif d <= -3.9e-7:
		tmp = t_1
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_2 + -1.0)
	elif (d <= 3.7e-87) or not (d <= 7.8e+37):
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	else:
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_1 = Float64(d * Float64(-(Float64(l * h) ^ -0.5)))
	t_2 = Float64(0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d))))
	tmp = 0.0
	if (d <= -1.05e+154)
		tmp = t_1;
	elseif (d <= -5.9e+93)
		tmp = Float64(t_0 * Float64(1.0 - t_2));
	elseif (d <= -3.9e-7)
		tmp = t_1;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_2 + -1.0));
	elseif ((d <= 3.7e-87) || !(d <= 7.8e+37))
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.125 * Float64(Float64(D * Float64(D * Float64(M * M))) / Float64(Float64(l * Float64(d * d)) / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	t_1 = d * -((l * h) ^ -0.5);
	t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	tmp = 0.0;
	if (d <= -1.05e+154)
		tmp = t_1;
	elseif (d <= -5.9e+93)
		tmp = t_0 * (1.0 - t_2);
	elseif (d <= -3.9e-7)
		tmp = t_1;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	elseif ((d <= 3.7e-87) || ~((d <= 7.8e+37)))
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	else
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.05e+154], t$95$1, If[LessEqual[d, -5.9e+93], N[(t$95$0 * N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.9e-7], t$95$1, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, 3.7e-87], N[Not[LessEqual[d, 7.8e+37]], $MachinePrecision]], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(0.125 * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.04999999999999997e154 or -5.90000000000000008e93 < d < -3.90000000000000025e-7

   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 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 64.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 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. associate-*r/ 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 73.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 54.7%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 54.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 54.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 54.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 45.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 45.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 45.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 65.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 65.8%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 65.8%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 65.8%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 65.6%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 65.6%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 65.6%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 65.6%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. unpow-1 65.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot \left(-d\right) \]

       5. metadata-eval 65.6%

          \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       6. pow-sqr 65.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \cdot \left(-d\right) \]

       7. rem-sqrt-square 67.6%

          \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       8. unpow1 67.6%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}}\right| \cdot \left(-d\right) \]

       9. sqr-pow 67.5%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(-d\right) \]

       10. fabs-sqr 67.5%

           \[\leadsto \color{blue}{\left({\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(-d\right) \]

       11. sqr-pow 67.6%

           \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}} \cdot \left(-d\right) \]

       12. unpow1 67.6%

           \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \left(-d\right) \]

       13. *-commutative 67.6%

           \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot \left(-d\right) \]

   15. Simplified 67.6%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.04999999999999997e154 < d < -5.90000000000000008e93

   1. Initial program 78.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 78.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 78.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 73.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 62.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 64.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 64.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 36.7%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. times-frac 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       2. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]

       3. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right) \]

       4. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right) \]

   12. Simplified 60.5%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]

   if -3.90000000000000025e-7 < d < -4.999999999999985e-310

   1. Initial program 51.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 51.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 51.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 53.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 53.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 53.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 53.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 53.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 40.3%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 40.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 40.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 40.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 40.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 40.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 40.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 40.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 59.9%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 59.9%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 59.9%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 59.9%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in M around 0 56.2%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   14. Step-by-step derivation

       1. times-frac 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       2. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]

       3. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right) \]

       4. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right) \]

   15. Simplified 54.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]

   if -4.999999999999985e-310 < d < 3.7000000000000002e-87 or 7.7999999999999997e37 < d

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 54.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 54.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 54.1%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 54.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 54.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 54.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 54.9%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 54.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 54.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 54.9%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 54.9%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 54.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 60.6%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 60.6%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   if 3.7000000000000002e-87 < d < 7.7999999999999997e37

   1. Initial program 84.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 84.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 84.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 93.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 93.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 93.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 93.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 93.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 93.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 93.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 93.1%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 93.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 93.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 93.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 93.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 93.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 84.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 84.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 84.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 68.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

       2. *-commutative 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

       3. associate-*r/ 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

       4. *-commutative 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

       5. associate-*r* 68.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell \cdot {d}^{2}}\right) \]

       6. associate-/l* 68.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}}\right) \]

       7. unpow2 68.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       8. associate-*l* 83.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       9. unpow2 83.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       10. unpow2 83.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \color{blue}{\left(d \cdot d\right)}}{h}}\right) \]

   12. Simplified 83.1%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+154}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -5.9 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-7}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) + -1\right)\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-87} \lor \neg \left(d \leq 7.8 \cdot 10^{+37}\right):\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \end{array} \]

Alternative 14: 52.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ t_2 := 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4 \cdot 10^{+92}:\\ \;\;\;\;t_0 \cdot \left(1 - t_2\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t_2 + -1\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h))))
        (t_1 (* d (- (pow (* l h) -0.5))))
        (t_2 (* 0.125 (* (/ (* D D) l) (/ (* h (* M M)) (* d d))))))
   (if (<= d -5.5e+153)
     t_1
     (if (<= d -4e+92)
       (* t_0 (- 1.0 t_2))
       (if (<= d -2e-5)
         t_1
         (if (<= d -5e-310)
           (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_2 -1.0))
           (if (<= d 2.05e-90)
             (* d (* (pow h -0.5) (pow l -0.5)))
             (if (<= d 3.6e+37)
               (*
                t_0
                (- 1.0 (* 0.125 (/ (* D (* D (* M M))) (/ (* l (* d d)) h)))))
               (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double t_1 = d * -pow((l * h), -0.5);
	double t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	double tmp;
	if (d <= -5.5e+153) {
		tmp = t_1;
	} else if (d <= -4e+92) {
		tmp = t_0 * (1.0 - t_2);
	} else if (d <= -2e-5) {
		tmp = t_1;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	} else if (d <= 2.05e-90) {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	} else if (d <= 3.6e+37) {
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    t_1 = d * -((l * h) ** (-0.5d0))
    t_2 = 0.125d0 * (((d_1 * d_1) / l) * ((h * (m * m)) / (d * d)))
    if (d <= (-5.5d+153)) then
        tmp = t_1
    else if (d <= (-4d+92)) then
        tmp = t_0 * (1.0d0 - t_2)
    else if (d <= (-2d-5)) then
        tmp = t_1
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_2 + (-1.0d0))
    else if (d <= 2.05d-90) then
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    else if (d <= 3.6d+37) then
        tmp = t_0 * (1.0d0 - (0.125d0 * ((d_1 * (d_1 * (m * m))) / ((l * (d * d)) / h))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    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) * (d / h)));
	double t_1 = d * -Math.pow((l * h), -0.5);
	double t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	double tmp;
	if (d <= -5.5e+153) {
		tmp = t_1;
	} else if (d <= -4e+92) {
		tmp = t_0 * (1.0 - t_2);
	} else if (d <= -2e-5) {
		tmp = t_1;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	} else if (d <= 2.05e-90) {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	} else if (d <= 3.6e+37) {
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	t_1 = d * -math.pow((l * h), -0.5)
	t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))
	tmp = 0
	if d <= -5.5e+153:
		tmp = t_1
	elif d <= -4e+92:
		tmp = t_0 * (1.0 - t_2)
	elif d <= -2e-5:
		tmp = t_1
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_2 + -1.0)
	elif d <= 2.05e-90:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	elif d <= 3.6e+37:
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_1 = Float64(d * Float64(-(Float64(l * h) ^ -0.5)))
	t_2 = Float64(0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d))))
	tmp = 0.0
	if (d <= -5.5e+153)
		tmp = t_1;
	elseif (d <= -4e+92)
		tmp = Float64(t_0 * Float64(1.0 - t_2));
	elseif (d <= -2e-5)
		tmp = t_1;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_2 + -1.0));
	elseif (d <= 2.05e-90)
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	elseif (d <= 3.6e+37)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.125 * Float64(Float64(D * Float64(D * Float64(M * M))) / Float64(Float64(l * Float64(d * d)) / h)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	t_1 = d * -((l * h) ^ -0.5);
	t_2 = 0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)));
	tmp = 0.0;
	if (d <= -5.5e+153)
		tmp = t_1;
	elseif (d <= -4e+92)
		tmp = t_0 * (1.0 - t_2);
	elseif (d <= -2e-5)
		tmp = t_1;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_2 + -1.0);
	elseif (d <= 2.05e-90)
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	elseif (d <= 3.6e+37)
		tmp = t_0 * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.5e+153], t$95$1, If[LessEqual[d, -4e+92], N[(t$95$0 * N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-5], t$95$1, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.05e-90], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e+37], N[(t$95$0 * N[(1.0 - N[(0.125 * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -5.5000000000000003e153 or -4.0000000000000002e92 < d < -2.00000000000000016e-5

   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 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 64.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 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. associate-*r/ 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 72.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 73.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 54.7%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 54.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 54.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 54.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 45.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 45.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 45.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 65.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 65.8%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 65.8%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 65.8%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 65.6%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 65.6%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 65.6%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 65.6%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. unpow-1 65.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot \left(-d\right) \]

       5. metadata-eval 65.6%

          \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       6. pow-sqr 65.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \cdot \left(-d\right) \]

       7. rem-sqrt-square 67.6%

          \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       8. unpow1 67.6%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}}\right| \cdot \left(-d\right) \]

       9. sqr-pow 67.5%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(-d\right) \]

       10. fabs-sqr 67.5%

           \[\leadsto \color{blue}{\left({\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(-d\right) \]

       11. sqr-pow 67.6%

           \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}} \cdot \left(-d\right) \]

       12. unpow1 67.6%

           \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \left(-d\right) \]

       13. *-commutative 67.6%

           \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot \left(-d\right) \]

   15. Simplified 67.6%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -5.5000000000000003e153 < d < -4.0000000000000002e92

   1. Initial program 78.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 78.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 78.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 78.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 73.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 62.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 64.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 64.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 36.7%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. times-frac 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       2. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]

       3. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right) \]

       4. unpow2 53.8%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right) \]

   12. Simplified 60.5%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]

   if -2.00000000000000016e-5 < d < -4.999999999999985e-310

   1. Initial program 51.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 51.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 51.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 52.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 51.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 53.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 53.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 53.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 53.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 53.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 40.3%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 40.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 40.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 40.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 40.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 40.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 40.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 40.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 59.9%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 59.9%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 59.9%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 59.9%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in M around 0 56.2%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   14. Step-by-step derivation

       1. times-frac 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       2. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]

       3. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right) \]

       4. unpow2 54.7%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right) \]

   15. Simplified 54.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]

   if -4.999999999999985e-310 < d < 2.05000000000000017e-90

   1. Initial program 41.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 41.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 41.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 41.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 41.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 41.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 41.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 41.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 41.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 30.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 30.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 30.0%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 30.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 30.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 30.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 30.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 29.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 29.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 30.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 30.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 31.4%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 31.4%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   if 2.05000000000000017e-90 < d < 3.59999999999999998e37

   1. Initial program 84.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 84.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 84.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 84.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 93.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 93.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 93.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 93.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 93.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 93.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 93.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 93.1%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 93.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 93.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 93.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 93.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 93.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 84.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 84.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 84.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 68.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

       2. *-commutative 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

       3. associate-*r/ 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

       4. *-commutative 68.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

       5. associate-*r* 68.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell \cdot {d}^{2}}\right) \]

       6. associate-/l* 68.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}}\right) \]

       7. unpow2 68.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       8. associate-*l* 83.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       9. unpow2 83.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       10. unpow2 83.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \color{blue}{\left(d \cdot d\right)}}{h}}\right) \]

   12. Simplified 83.1%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}}\right) \]

   if 3.59999999999999998e37 < d

   1. Initial program 80.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 72.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 72.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 72.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 72.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 72.2%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 72.3%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 72.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. sqrt-div 82.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]

   10. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]
3. Recombined 6 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+153}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-5}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) + -1\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 15: 47.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.5 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) + -1\right)\\ \mathbf{elif}\;h \leq 2.8 \cdot 10^{-303}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;h \leq 2.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.5e-238)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ (* 0.125 (* (/ (* D D) l) (/ (* h (* M M)) (* d d)))) -1.0))
   (if (<= h 2.8e-303)
     (* d (- (pow (* l h) -0.5)))
     (if (<= h 2.3e+37)
       (/ d (sqrt (* l h)))
       (*
        (sqrt (* (/ d l) (/ d h)))
        (- 1.0 (* 0.125 (/ (* D (* D (* M M))) (/ (* l (* d d)) h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.5e-238) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))) + -1.0);
	} else if (h <= 2.8e-303) {
		tmp = d * -pow((l * h), -0.5);
	} else if (h <= 2.3e+37) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * 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 (h <= (-1.5d-238)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.125d0 * (((d_1 * d_1) / l) * ((h * (m * m)) / (d * d)))) + (-1.0d0))
    else if (h <= 2.8d-303) then
        tmp = d * -((l * h) ** (-0.5d0))
    else if (h <= 2.3d+37) then
        tmp = d / sqrt((l * h))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * ((d_1 * (d_1 * (m * m))) / ((l * (d * 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 (h <= -1.5e-238) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))) + -1.0);
	} else if (h <= 2.8e-303) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else if (h <= 2.3e+37) {
		tmp = d / Math.sqrt((l * h));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1.5e-238:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))) + -1.0)
	elif h <= 2.8e-303:
		tmp = d * -math.pow((l * h), -0.5)
	elif h <= 2.3e+37:
		tmp = d / math.sqrt((l * h))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.5e-238)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d)))) + -1.0));
	elseif (h <= 2.8e-303)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	elseif (h <= 2.3e+37)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D * Float64(D * Float64(M * M))) / Float64(Float64(l * Float64(d * d)) / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1.5e-238)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))) + -1.0);
	elseif (h <= 2.8e-303)
		tmp = d * -((l * h) ^ -0.5);
	elseif (h <= 2.3e+37)
		tmp = d / sqrt((l * h));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.5e-238], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.8e-303], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[h, 2.3e+37], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.5e-238

   1. Initial program 62.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 62.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 62.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 62.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 62.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 62.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 62.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 62.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 62.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 62.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 62.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 62.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 64.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 64.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 64.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 65.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 65.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 65.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 65.5%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 50.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 50.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 50.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 50.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 46.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 46.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 46.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 60.5%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 60.5%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 60.5%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 60.5%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in M around 0 50.0%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   14. Step-by-step derivation

       1. times-frac 49.1%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       2. unpow2 49.1%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]

       3. unpow2 49.1%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right) \]

       4. unpow2 49.1%

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right) \]

   15. Simplified 49.1%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]

   if -1.5e-238 < h < 2.8e-303

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 47.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 48.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 48.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 48.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 42.4%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 42.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 42.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 42.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 41.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 41.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 41.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 63.2%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 63.2%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 63.2%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 63.2%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 75.2%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 75.2%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 75.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 75.2%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. unpow-1 75.2%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot \left(-d\right) \]

       5. metadata-eval 75.2%

          \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       6. pow-sqr 75.1%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \cdot \left(-d\right) \]

       7. rem-sqrt-square 80.9%

          \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       8. unpow1 80.9%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}}\right| \cdot \left(-d\right) \]

       9. sqr-pow 80.9%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(-d\right) \]

       10. fabs-sqr 80.9%

           \[\leadsto \color{blue}{\left({\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(-d\right) \]

       11. sqr-pow 80.9%

           \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}} \cdot \left(-d\right) \]

       12. unpow1 80.9%

           \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \left(-d\right) \]

       13. *-commutative 80.9%

           \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot \left(-d\right) \]

   15. Simplified 80.9%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 2.8e-303 < h < 2.30000000000000002e37

   1. Initial program 73.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 74.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 65.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 65.9%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 65.9%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 65.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 65.9%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 65.9%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 65.9%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. pow1 65.9%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right)}^{1}} \]

      2. *-commutative 65.9%

         \[\leadsto {\color{blue}{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}}^{1} \]

      3. associate-/r* 65.9%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)}^{1} \]

      4. sqrt-div 67.0%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)}^{1} \]

      5. metadata-eval 67.0%

         \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)}^{1} \]

   10. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 67.0%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 67.0%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 67.0%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

       4. *-commutative 67.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Simplified 67.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   if 2.30000000000000002e37 < h

   1. Initial program 56.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 56.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 56.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 60.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 60.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 60.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 60.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 60.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 60.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 60.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 60.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 56.4%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 56.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 56.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 56.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 53.1%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 53.1%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 53.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 38.3%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

       2. *-commutative 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

       3. associate-*r/ 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

       4. *-commutative 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

       5. associate-*r* 39.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell \cdot {d}^{2}}\right) \]

       6. associate-/l* 38.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}}\right) \]

       7. unpow2 38.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       8. associate-*l* 43.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       9. unpow2 43.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       10. unpow2 43.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \color{blue}{\left(d \cdot d\right)}}{h}}\right) \]

   12. Simplified 43.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.5 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) + -1\right)\\ \mathbf{elif}\;h \leq 2.8 \cdot 10^{-303}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;h \leq 2.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \end{array} \]

Alternative 16: 48.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-33}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{d}\right) \cdot \frac{h}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.55e-33)
   (* d (- (sqrt (/ 1.0 (* l h)))))
   (if (<= l 6.2e-233)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (- 1.0 (* 0.125 (* (* (* M M) (/ (* D D) d)) (/ h (* l d))))))
     (/ d (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.55e-33) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else if (l <= 6.2e-233) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * (((M * M) * ((D * D) / d)) * (h / (l * d)))));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.55d-33)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else if (l <= 6.2d-233) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * (((m * m) * ((d_1 * d_1) / d)) * (h / (l * d)))))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.55e-33) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else if (l <= 6.2e-233) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * (((M * M) * ((D * D) / d)) * (h / (l * d)))));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.55e-33:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	elif l <= 6.2e-233:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * (((M * M) * ((D * D) / d)) * (h / (l * d)))))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.55e-33)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	elseif (l <= 6.2e-233)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(M * M) * Float64(Float64(D * D) / d)) * Float64(h / Float64(l * d))))));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.55e-33)
		tmp = d * -sqrt((1.0 / (l * h)));
	elseif (l <= 6.2e-233)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * (((M * M) * ((D * D) / d)) * (h / (l * d)))));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.55e-33], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 6.2e-233], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(N[(M * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.54999999999999998e-33

   1. Initial program 54.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 54.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 54.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 54.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 54.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 54.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 52.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 52.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 52.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 52.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 52.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 52.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 52.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 52.8%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 33.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 33.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 33.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 33.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 33.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 33.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 33.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 56.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 56.8%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 56.8%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 56.8%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 35.4%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 35.4%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. neg-mul-1 35.4%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

       3. associate-/r* 35.3%

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       4. associate-/l/ 35.4%

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   15. Simplified 35.4%

       \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if -1.54999999999999998e-33 < l < 6.2000000000000003e-233

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 76.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 76.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 76.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 76.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 76.1%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 69.1%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 69.1%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 69.1%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 69.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 63.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 63.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 63.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 52.6%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 52.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

       2. *-commutative 52.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

       3. associate-*r/ 52.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

       4. *-commutative 52.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

       5. associate-*r* 52.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell \cdot {d}^{2}}\right) \]

       6. *-commutative 52.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

       7. unpow2 52.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

       8. associate-*l* 56.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

       9. times-frac 54.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d \cdot \ell}\right)}\right) \]

       10. associate-/l* 51.5%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d \cdot \ell}\right)\right) \]

       11. associate-/r/ 54.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \frac{h}{d \cdot \ell}\right)\right) \]

       12. unpow2 54.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \frac{h}{d \cdot \ell}\right)\right) \]

       13. unpow2 54.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D \cdot D}{d} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{h}{d \cdot \ell}\right)\right) \]

   12. Simplified 54.1%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot \ell}\right)}\right) \]

   if 6.2000000000000003e-233 < l

   1. Initial program 64.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 64.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 64.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 64.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 64.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 64.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 64.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 64.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 54.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 54.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 54.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 54.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 54.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 54.2%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 54.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. pow1 54.2%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right)}^{1}} \]

      2. *-commutative 54.2%

         \[\leadsto {\color{blue}{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}}^{1} \]

      3. associate-/r* 54.2%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)}^{1} \]

      4. sqrt-div 54.9%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)}^{1} \]

      5. metadata-eval 54.9%

         \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)}^{1} \]

   10. Applied egg-rr 54.9%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 54.9%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 54.9%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 54.9%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

       4. *-commutative 54.9%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Simplified 54.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-33}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{d}\right) \cdot \frac{h}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 17: 47.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{-33}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5.4e-33)
   (* d (- (sqrt (/ 1.0 (* l h)))))
   (if (<= l 1.6e-150)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (- 1.0 (* 0.125 (/ (* D (* D (* M M))) (/ (* l (* d d)) h)))))
     (/ d (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.4e-33) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else if (l <= 1.6e-150) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5.4d-33)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else if (l <= 1.6d-150) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * ((d_1 * (d_1 * (m * m))) / ((l * (d * d)) / h))))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.4e-33) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else if (l <= 1.6e-150) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -5.4e-33:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	elif l <= 1.6e-150:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5.4e-33)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	elseif (l <= 1.6e-150)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D * Float64(D * Float64(M * M))) / Float64(Float64(l * Float64(d * d)) / h)))));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -5.4e-33)
		tmp = d * -sqrt((1.0 / (l * h)));
	elseif (l <= 1.6e-150)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D * (D * (M * M))) / ((l * (d * d)) / h))));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5.4e-33], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 1.6e-150], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.4000000000000002e-33

   1. Initial program 54.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 54.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 54.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 54.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 54.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 54.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 54.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 52.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 52.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 52.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 52.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 52.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 52.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 52.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 52.8%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 33.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 33.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 33.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 33.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 33.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 33.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 33.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 56.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 56.8%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 56.8%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 56.8%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 35.4%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 35.4%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. neg-mul-1 35.4%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

       3. associate-/r* 35.3%

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       4. associate-/l/ 35.4%

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   15. Simplified 35.4%

       \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if -5.4000000000000002e-33 < l < 1.5999999999999999e-150

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 76.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 77.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 77.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 77.6%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 69.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 69.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 69.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 69.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 62.1%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 62.1%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 62.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in M around 0 51.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 51.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

       2. *-commutative 51.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

       3. associate-*r/ 51.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

       4. *-commutative 51.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

       5. associate-*r* 52.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell \cdot {d}^{2}}\right) \]

       6. associate-/l* 50.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}}\right) \]

       7. unpow2 50.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       8. associate-*l* 54.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       9. unpow2 54.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)}{\frac{\ell \cdot {d}^{2}}{h}}\right) \]

       10. unpow2 54.2%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \color{blue}{\left(d \cdot d\right)}}{h}}\right) \]

   12. Simplified 54.2%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}}\right) \]

   if 1.5999999999999999e-150 < l

   1. Initial program 63.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 56.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 56.3%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 56.3%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 56.3%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 56.3%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 56.3%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 56.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. pow1 56.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right)}^{1}} \]

      2. *-commutative 56.3%

         \[\leadsto {\color{blue}{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}}^{1} \]

      3. associate-/r* 56.3%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)}^{1} \]

      4. sqrt-div 56.4%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)}^{1} \]

      5. metadata-eval 56.4%

         \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)}^{1} \]

   10. Applied egg-rr 56.4%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 56.4%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 56.5%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 56.5%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

       4. *-commutative 56.5%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Simplified 56.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{-33}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{\ell \cdot \left(d \cdot d\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 18: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{-33}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-300}:\\ \;\;\;\;d \cdot {\left(\ell \cdot \left(\ell \cdot \left(h \cdot h\right)\right)\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.32e-33)
   (* d (- (pow (* l h) -0.5)))
   (if (<= d -2.7e-300)
     (* d (pow (* l (* l (* h h))) -0.25))
     (/ d (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.32e-33) {
		tmp = d * -pow((l * h), -0.5);
	} else if (d <= -2.7e-300) {
		tmp = d * pow((l * (l * (h * h))), -0.25);
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-1.32d-33)) then
        tmp = d * -((l * h) ** (-0.5d0))
    else if (d <= (-2.7d-300)) then
        tmp = d * ((l * (l * (h * h))) ** (-0.25d0))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.32e-33) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else if (d <= -2.7e-300) {
		tmp = d * Math.pow((l * (l * (h * h))), -0.25);
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.32e-33:
		tmp = d * -math.pow((l * h), -0.5)
	elif d <= -2.7e-300:
		tmp = d * math.pow((l * (l * (h * h))), -0.25)
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.32e-33)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	elseif (d <= -2.7e-300)
		tmp = Float64(d * (Float64(l * Float64(l * Float64(h * h))) ^ -0.25));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.32e-33)
		tmp = d * -((l * h) ^ -0.5);
	elseif (d <= -2.7e-300)
		tmp = d * ((l * (l * (h * h))) ^ -0.25);
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.32e-33], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -2.7e-300], N[(d * N[Power[N[(l * N[(l * N[(h * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.31999999999999993e-33

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 74.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 74.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 74.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 74.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 74.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 74.6%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 58.0%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 58.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 58.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 58.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 52.5%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 52.5%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 52.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 65.5%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 65.5%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 65.5%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 65.5%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 54.8%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 54.8%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 54.8%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 54.8%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. unpow-1 54.8%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot \left(-d\right) \]

       5. metadata-eval 54.8%

          \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       6. pow-sqr 54.8%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \cdot \left(-d\right) \]

       7. rem-sqrt-square 56.2%

          \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       8. unpow1 56.2%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}}\right| \cdot \left(-d\right) \]

       9. sqr-pow 56.1%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(-d\right) \]

       10. fabs-sqr 56.1%

           \[\leadsto \color{blue}{\left({\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(-d\right) \]

       11. sqr-pow 56.2%

           \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}} \cdot \left(-d\right) \]

       12. unpow1 56.2%

           \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \left(-d\right) \]

       13. *-commutative 56.2%

           \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot \left(-d\right) \]

   15. Simplified 56.2%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.31999999999999993e-33 < d < -2.69999999999999995e-300

   1. Initial program 48.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 48.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 48.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 48.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 48.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 48.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 48.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 13.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 13.5%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 13.5%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 13.5%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 13.5%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 13.5%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 11.8%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 11.8%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 11.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 11.8%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 11.8%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 11.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. sqr-pow 11.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      2. *-commutative 11.8%

         \[\leadsto \left({\color{blue}{\left(\ell \cdot h\right)}}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right) \cdot d \]

      3. metadata-eval 11.8%

         \[\leadsto \left({\left(\ell \cdot h\right)}^{\color{blue}{-0.25}} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right) \cdot d \]

      4. *-commutative 11.8%

         \[\leadsto \left({\left(\ell \cdot h\right)}^{-0.25} \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{\left(\frac{-0.5}{2}\right)}\right) \cdot d \]

      5. metadata-eval 11.8%

         \[\leadsto \left({\left(\ell \cdot h\right)}^{-0.25} \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.25}}\right) \cdot d \]

   10. Applied egg-rr 11.8%

       \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{-0.25} \cdot {\left(\ell \cdot h\right)}^{-0.25}\right)} \cdot d \]
   11. Step-by-step derivation

       1. pow-prod-down 22.7%

          \[\leadsto \color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{-0.25}} \cdot d \]

   12. Applied egg-rr 22.7%

       \[\leadsto \color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{-0.25}} \cdot d \]
   13. Step-by-step derivation

       1. associate-*l* 22.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \left(h \cdot \left(\ell \cdot h\right)\right)\right)}}^{-0.25} \cdot d \]

       2. *-commutative 22.6%

          \[\leadsto {\left(\ell \cdot \left(h \cdot \color{blue}{\left(h \cdot \ell\right)}\right)\right)}^{-0.25} \cdot d \]

       3. associate-*r* 27.8%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left(\left(h \cdot h\right) \cdot \ell\right)}\right)}^{-0.25} \cdot d \]

   14. Simplified 27.8%

       \[\leadsto \color{blue}{{\left(\ell \cdot \left(\left(h \cdot h\right) \cdot \ell\right)\right)}^{-0.25}} \cdot d \]

   if -2.69999999999999995e-300 < d

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 66.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 66.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 66.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 66.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 66.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 66.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 51.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 51.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 51.0%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 51.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. *-commutative 51.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      3. associate-/r* 51.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   8. Simplified 51.0%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
   9. Step-by-step derivation

      1. pow1 51.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right)}^{1}} \]

      2. *-commutative 51.0%

         \[\leadsto {\color{blue}{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}}^{1} \]

      3. associate-/r* 51.0%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)}^{1} \]

      4. sqrt-div 51.6%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)}^{1} \]

      5. metadata-eval 51.6%

         \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)}^{1} \]

   10. Applied egg-rr 51.6%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 51.6%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 51.7%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 51.7%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

       4. *-commutative 51.7%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Simplified 51.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{-33}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-300}:\\ \;\;\;\;d \cdot {\left(\ell \cdot \left(\ell \cdot \left(h \cdot h\right)\right)\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 19: 41.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -7.4e-37)
   (* d (- (pow (* l h) -0.5)))
   (* d (sqrt (/ 1.0 (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -7.4e-37) {
		tmp = d * -pow((l * h), -0.5);
	} 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 (d <= (-7.4d-37)) then
        tmp = d * -((l * h) ** (-0.5d0))
    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 (d <= -7.4e-37) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -7.4e-37:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -7.4e-37)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -7.4e-37)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -7.4e-37], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.4e-37

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 74.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 74.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 74.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 74.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 74.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 74.6%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. sqrt-unprod 58.0%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1} \]

      3. associate-*l* 58.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right)\right)}^{1} \]

      4. associate-*l* 58.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   7. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 58.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

      2. associate-/l* 52.5%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right) \]

      3. *-commutative 52.5%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{\color{blue}{h \cdot 0.5}}}\right) \]

   9. Simplified 52.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right)} \]

   10. Taylor expanded in d around -inf 65.5%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 65.5%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

       2. mul-1-neg 65.5%

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   12. Simplified 65.5%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot 0.5}}\right) \]

   13. Taylor expanded in d around inf 54.8%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 54.8%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 54.8%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 54.8%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. unpow-1 54.8%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot \left(-d\right) \]

       5. metadata-eval 54.8%

          \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       6. pow-sqr 54.8%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \cdot \left(-d\right) \]

       7. rem-sqrt-square 56.2%

          \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       8. unpow1 56.2%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}}\right| \cdot \left(-d\right) \]

       9. sqr-pow 56.1%

          \[\leadsto \left|\color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(-d\right) \]

       10. fabs-sqr 56.1%

           \[\leadsto \color{blue}{\left({\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(-d\right) \]

       11. sqr-pow 56.2%

           \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-0.5}\right)}^{1}} \cdot \left(-d\right) \]

       12. unpow1 56.2%

           \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \left(-d\right) \]

       13. *-commutative 56.2%

           \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot \left(-d\right) \]

   15. Simplified 56.2%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -7.4e-37 < d

   1. Initial program 61.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 40.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \]

Alternative 20: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\frac{1}{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ 1.0 (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt((1.0 / (l * h)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt((1.0d0 / (l * h)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt((1.0 / (l * h)));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt((1.0 / (l * h)))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt((1.0 / (l * h)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 63.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 63.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 63.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 30.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

5. Final simplification 30.6%

   \[\leadsto d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

Alternative 21: 26.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / l) / h))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 63.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 63.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 63.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 30.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-un-lft-identity 30.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   2. *-commutative 30.6%

      \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

6. Applied egg-rr 30.6%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
7. Step-by-step derivation

   1. *-lft-identity 30.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   2. *-commutative 30.6%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   3. associate-/r* 30.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

8. Simplified 30.6%

   \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

9. Final simplification 30.6%

   \[\leadsto d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]

Alternative 22: 26.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 63.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 63.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 63.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 30.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-un-lft-identity 30.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   2. *-commutative 30.6%

      \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

6. Applied egg-rr 30.6%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
7. Step-by-step derivation

   1. *-lft-identity 30.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   2. *-commutative 30.6%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   3. associate-/r* 30.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

8. Simplified 30.6%

   \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
9. Step-by-step derivation

   1. pow1 30.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right)}^{1}} \]

   2. *-commutative 30.6%

      \[\leadsto {\color{blue}{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}}^{1} \]

   3. associate-/r* 30.6%

      \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)}^{1} \]

   4. sqrt-div 30.5%

      \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)}^{1} \]

   5. metadata-eval 30.5%

      \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)}^{1} \]

10. Applied egg-rr 30.5%

    \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
11. Step-by-step derivation

    1. unpow1 30.5%

       \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

    2. associate-*r/ 30.5%

       \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

    3. *-rgt-identity 30.5%

       \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

    4. *-commutative 30.5%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

12. Simplified 30.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

13. Final simplification 30.5%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

Reproduce

herbie shell --seed 2023228 
(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)))))