Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 79.9%

Time: 20.6s

Alternatives: 27

Speedup: 3.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{0.5}{d}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot t\_0}{\ell}\right) \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\frac{-1}{h}} \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot t\_0\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ 0.5 d))) (t_1 (sqrt (- d))))
   (if (<= h -2.25e-61)
     (*
      (- 1.0 (* (/ (* (/ M d) (* (* D 0.5) 0.5)) (/ 1.0 h)) (/ (* D t_0) l)))
      (* (/ t_1 (sqrt (- l))) (sqrt (/ d h))))
     (if (<= h -1e-310)
       (*
        (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
        (* (pow (/ d l) (/ 1.0 2.0)) (* (sqrt (/ -1.0 h)) t_1)))
       (*
        (- 1.0 (* (/ D l) (* (* (* 0.25 D) (* (/ M d) h)) t_0)))
        (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (0.5 / d);
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -2.25e-61) {
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / sqrt(-l)) * sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * (sqrt((-1.0 / h)) * t_1));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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) :: tmp
    t_0 = m * (0.5d0 / d)
    t_1 = sqrt(-d)
    if (h <= (-2.25d-61)) then
        tmp = (1.0d0 - ((((m / d) * ((d_1 * 0.5d0) * 0.5d0)) / (1.0d0 / h)) * ((d_1 * t_0) / l))) * ((t_1 / sqrt(-l)) * sqrt((d / h)))
    else if (h <= (-1d-310)) then
        tmp = (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * (sqrt(((-1.0d0) / h)) * t_1))
    else
        tmp = (1.0d0 - ((d_1 / l) * (((0.25d0 * d_1) * ((m / d) * h)) * t_0))) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = M * (0.5 / d);
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (h <= -2.25e-61) {
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / Math.sqrt(-l)) * Math.sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * (Math.sqrt((-1.0 / h)) * t_1));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (0.5 / d)
	t_1 = math.sqrt(-d)
	tmp = 0
	if h <= -2.25e-61:
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / math.sqrt(-l)) * math.sqrt((d / h)))
	elif h <= -1e-310:
		tmp = (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * (math.sqrt((-1.0 / h)) * t_1))
	else:
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(0.5 / d))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2.25e-61)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / Float64(1.0 / h)) * Float64(Float64(D * t_0) / l))) * Float64(Float64(t_1 / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-1.0 / h)) * t_1)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(D / l) * Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) * t_0))) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (0.5 / d);
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (h <= -2.25e-61)
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / sqrt(-l)) * sqrt((d / h)));
	elseif (h <= -1e-310)
		tmp = (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * (sqrt((-1.0 / h)) * t_1));
	else
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2.25e-61], N[(N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(D / l), $MachinePrecision] * N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.25e-61

   1. Initial program 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 80.2

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

   8. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 91.5

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

   10. Applied rewrites 91.5%

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

   if -2.25e-61 < h < -9.999999999999969e-311

   1. Initial program 73.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   if -9.999999999999969e-311 < h

   1. Initial program 60.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 71.9

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

   4. Applied rewrites 71.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.9

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. unpow-prod-down N/A

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

      8. pow1/2 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 74.2

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

   6. Applied rewrites 74.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. unpow-prod-down N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. pow2 N/A

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

      11. swap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

      18. associate-/l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*l* N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. associate-*r* N/A

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

   8. Applied rewrites 75.6%

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

   10. Applied rewrites 83.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\frac{-1}{h}} \cdot \sqrt{-d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\left(-0.125 \cdot h\right) \cdot \left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot \left(D \cdot M\right)\right)\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -4e-53)
     (* (* (* (* -0.125 h) (* (* (/ D (* (* d d) l)) M) (* D M))) t_1) t_2)
     (if (<= t_0 2e+282) (* t_2 t_1) (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-53) {
		tmp = (((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) * t_1) * t_2;
	} else if (t_0 <= 2e+282) {
		tmp = t_2 * t_1;
	} else {
		tmp = d / (sqrt(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 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-4d-53)) then
        tmp = ((((-0.125d0) * h) * (((d_1 / ((d * d) * l)) * m) * (d_1 * m))) * t_1) * t_2
    else if (t_0 <= 2d+282) then
        tmp = t_2 * t_1
    else
        tmp = d / (sqrt(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.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-53) {
		tmp = (((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) * t_1) * t_2;
	} else if (t_0 <= 2e+282) {
		tmp = t_2 * t_1;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -4e-53:
		tmp = (((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) * t_1) * t_2
	elif t_0 <= 2e+282:
		tmp = t_2 * t_1
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -4e-53)
		tmp = Float64(Float64(Float64(Float64(-0.125 * h) * Float64(Float64(Float64(D / Float64(Float64(d * d) * l)) * M) * Float64(D * M))) * t_1) * t_2);
	elseif (t_0 <= 2e+282)
		tmp = Float64(t_2 * t_1);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -4e-53)
		tmp = (((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) * t_1) * t_2;
	elseif (t_0 <= 2e+282)
		tmp = t_2 * t_1;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-53], N[(N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(N[(N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], N[(t$95$2 * t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.00000000000000012e-53

   1. Initial program 88.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 37.7

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in h around inf

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

      1. associate-*r* N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

   9. Applied rewrites 55.6%

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

   if -4.00000000000000012e-53 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 89.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 86.3%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.3%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\left(-0.125 \cdot h\right) \cdot \left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-0.125 \cdot h\right) \cdot \left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot \left(D \cdot M\right)\right)}{\sqrt{\frac{h}{\frac{d}{\ell} \cdot d}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))))
   (if (<= t_0 -4e-53)
     (/
      (* (* -0.125 h) (* (* (/ D (* (* d d) l)) M) (* D M)))
      (sqrt (/ h (* (/ d l) d))))
     (if (<= t_0 2e+282)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -4e-53) {
		tmp = ((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) / sqrt((h / ((d / l) * d)));
	} else if (t_0 <= 2e+282) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(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) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    if (t_0 <= (-4d-53)) then
        tmp = (((-0.125d0) * h) * (((d_1 / ((d * d) * l)) * m) * (d_1 * m))) / sqrt((h / ((d / l) * d)))
    else if (t_0 <= 2d+282) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = d / (sqrt(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.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -4e-53) {
		tmp = ((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) / Math.sqrt((h / ((d / l) * d)));
	} else if (t_0 <= 2e+282) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	tmp = 0
	if t_0 <= -4e-53:
		tmp = ((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) / math.sqrt((h / ((d / l) * d)))
	elif t_0 <= 2e+282:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -4e-53)
		tmp = Float64(Float64(Float64(-0.125 * h) * Float64(Float64(Float64(D / Float64(Float64(d * d) * l)) * M) * Float64(D * M))) / sqrt(Float64(h / Float64(Float64(d / l) * d))));
	elseif (t_0 <= 2e+282)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -4e-53)
		tmp = ((-0.125 * h) * (((D / ((d * d) * l)) * M) * (D * M))) / sqrt((h / ((d / l) * d)));
	elseif (t_0 <= 2e+282)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-53], N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(N[(N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / N[(N[(d / l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.00000000000000012e-53

   1. Initial program 88.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 37.7

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in h around inf

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

      1. associate-*r* N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

   9. Applied rewrites 44.8%

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

   if -4.00000000000000012e-53 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 89.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 86.3%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.3%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-0.125 \cdot h\right) \cdot \left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot \left(D \cdot M\right)\right)}{\sqrt{\frac{h}{\frac{d}{\ell} \cdot d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}} \cdot \left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))))
   (if (<= t_0 -2e+73)
     (*
      (sqrt (/ (* d d) (* l h)))
      (- 1.0 (* (* (* (* 0.125 (/ D (* (* d d) l))) h) M) (* D M))))
     (if (<= t_0 2e+282)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -2e+73) {
		tmp = sqrt(((d * d) / (l * h))) * (1.0 - ((((0.125 * (D / ((d * d) * l))) * h) * M) * (D * M)));
	} else if (t_0 <= 2e+282) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(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) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    if (t_0 <= (-2d+73)) then
        tmp = sqrt(((d * d) / (l * h))) * (1.0d0 - ((((0.125d0 * (d_1 / ((d * d) * l))) * h) * m) * (d_1 * m)))
    else if (t_0 <= 2d+282) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = d / (sqrt(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.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -2e+73) {
		tmp = Math.sqrt(((d * d) / (l * h))) * (1.0 - ((((0.125 * (D / ((d * d) * l))) * h) * M) * (D * M)));
	} else if (t_0 <= 2e+282) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	tmp = 0
	if t_0 <= -2e+73:
		tmp = math.sqrt(((d * d) / (l * h))) * (1.0 - ((((0.125 * (D / ((d * d) * l))) * h) * M) * (D * M)))
	elif t_0 <= 2e+282:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -2e+73)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(0.125 * Float64(D / Float64(Float64(d * d) * l))) * h) * M) * Float64(D * M))));
	elseif (t_0 <= 2e+282)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -2e+73)
		tmp = sqrt(((d * d) / (l * h))) * (1.0 - ((((0.125 * (D / ((d * d) * l))) * h) * M) * (D * M)));
	elseif (t_0 <= 2e+282)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+73], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(0.125 * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999997e73

   1. Initial program 88.4%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 40.9%

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

   if -1.99999999999999997e73 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 90.2%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.0

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

   8. Applied rewrites 36.0%

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

   10. Applied rewrites 82.6%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -2 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}} \cdot \left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \left(-0.125 \cdot h\right)\right) \cdot \sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))))
   (if (<= t_0 -4e-53)
     (*
      (* (* (* (* M M) D) (/ D (* (* d d) l))) (* -0.125 h))
      (sqrt (/ (* d d) (* l h))))
     (if (<= t_0 2e+282)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -4e-53) {
		tmp = ((((M * M) * D) * (D / ((d * d) * l))) * (-0.125 * h)) * sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 2e+282) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(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) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    if (t_0 <= (-4d-53)) then
        tmp = ((((m * m) * d_1) * (d_1 / ((d * d) * l))) * ((-0.125d0) * h)) * sqrt(((d * d) / (l * h)))
    else if (t_0 <= 2d+282) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = d / (sqrt(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.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -4e-53) {
		tmp = ((((M * M) * D) * (D / ((d * d) * l))) * (-0.125 * h)) * Math.sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 2e+282) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	tmp = 0
	if t_0 <= -4e-53:
		tmp = ((((M * M) * D) * (D / ((d * d) * l))) * (-0.125 * h)) * math.sqrt(((d * d) / (l * h)))
	elif t_0 <= 2e+282:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -4e-53)
		tmp = Float64(Float64(Float64(Float64(Float64(M * M) * D) * Float64(D / Float64(Float64(d * d) * l))) * Float64(-0.125 * h)) * sqrt(Float64(Float64(d * d) / Float64(l * h))));
	elseif (t_0 <= 2e+282)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -4e-53)
		tmp = ((((M * M) * D) * (D / ((d * d) * l))) * (-0.125 * h)) * sqrt(((d * d) / (l * h)));
	elseif (t_0 <= 2e+282)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-53], N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * h), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.00000000000000012e-53

   1. Initial program 88.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 37.7

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in h around inf

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

      1. associate-*r* N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-times N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 38.8

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

   9. Applied rewrites 38.8%

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

   if -4.00000000000000012e-53 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 89.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 86.3%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.3%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \left(-0.125 \cdot h\right)\right) \cdot \sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+280}:\\ \;\;\;\;\left(\frac{D \cdot D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))))
   (if (<= t_0 -4e+280)
     (* (* (/ (* D D) d) (* (* M M) 0.125)) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 2e+282)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -4e+280) {
		tmp = (((D * D) / d) * ((M * M) * 0.125)) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 2e+282) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(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) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    if (t_0 <= (-4d+280)) then
        tmp = (((d_1 * d_1) / d) * ((m * m) * 0.125d0)) * sqrt((h / ((l * l) * l)))
    else if (t_0 <= 2d+282) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = d / (sqrt(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.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -4e+280) {
		tmp = (((D * D) / d) * ((M * M) * 0.125)) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 2e+282) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	tmp = 0
	if t_0 <= -4e+280:
		tmp = (((D * D) / d) * ((M * M) * 0.125)) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= 2e+282:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -4e+280)
		tmp = Float64(Float64(Float64(Float64(D * D) / d) * Float64(Float64(M * M) * 0.125)) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= 2e+282)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -4e+280)
		tmp = (((D * D) / d) * ((M * M) * 0.125)) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= 2e+282)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+280], N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.0000000000000001e280

   1. Initial program 87.6%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-/l* N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

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

   if -4.0000000000000001e280 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 90.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.0

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

   8. Applied rewrites 34.0%

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

   10. Applied rewrites 77.5%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -4 \cdot 10^{+280}:\\ \;\;\;\;\left(\frac{D \cdot D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\left(-t\_0\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))))
   (if (<= t_2 0.0)
     (* (- t_0) t_1)
     (if (<= t_2 2e+282) (* t_0 t_1) (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -t_0 * t_1;
	} else if (t_2 <= 2e+282) {
		tmp = t_0 * t_1;
	} else {
		tmp = d / (sqrt(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))
    t_1 = sqrt((d / h))
    t_2 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    if (t_2 <= 0.0d0) then
        tmp = -t_0 * t_1
    else if (t_2 <= 2d+282) then
        tmp = t_0 * t_1
    else
        tmp = d / (sqrt(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));
	double t_1 = Math.sqrt((d / h));
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -t_0 * t_1;
	} else if (t_2 <= 2e+282) {
		tmp = t_0 * t_1;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = -t_0 * t_1
	elif t_2 <= 2e+282:
		tmp = t_0 * t_1
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(-t_0) * t_1);
	elseif (t_2 <= 2e+282)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = -t_0 * t_1;
	elseif (t_2 <= 2e+282)
		tmp = t_0 * t_1;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[((-t$95$0) * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+282], N[(t$95$0 * t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.6%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 8.9

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

   10. Applied rewrites 8.9%

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

       1. lift-/.f64 N/A

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

   12. Applied rewrites 24.7%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 98.2%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.1

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

   8. Applied rewrites 40.1%

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

   10. Applied rewrites 96.7%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.3%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 0:\\ \;\;\;\;\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(1 - \frac{\left(0.25 \cdot \frac{M}{d}\right) \cdot D}{\frac{\ell}{h}} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 - \left(\left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot D\right) \cdot M\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
      2e+282)
   (*
    (- 1.0 (* (/ (* (* 0.25 (/ M d)) D) (/ l h)) (* D (* M (/ 0.5 d)))))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (/
    (/
     (*
      (*
       (- 1.0 (* (* (* (* (* 0.125 (/ D (* (* d d) l))) h) M) D) M))
       (sqrt d))
      (sqrt d))
     (sqrt l))
    (sqrt h))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282) {
		tmp = (1.0 - ((((0.25 * (M / d)) * D) / (l / h)) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * sqrt(d)) * sqrt(d)) / sqrt(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 / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))) <= 2d+282) then
        tmp = (1.0d0 - ((((0.25d0 * (m / d)) * d_1) / (l / h)) * (d_1 * (m * (0.5d0 / d))))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = ((((1.0d0 - (((((0.125d0 * (d_1 / ((d * d) * l))) * h) * m) * d_1) * m)) * sqrt(d)) * sqrt(d)) / sqrt(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 (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282) {
		tmp = (1.0 - ((((0.25 * (M / d)) * D) / (l / h)) * (D * (M * (0.5 / d))))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * Math.sqrt(d)) * Math.sqrt(d)) / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282:
		tmp = (1.0 - ((((0.25 * (M / d)) * D) / (l / h)) * (D * (M * (0.5 / d))))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * math.sqrt(d)) * math.sqrt(d)) / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0))))) <= 2e+282)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * Float64(M / d)) * D) / Float64(l / h)) * Float64(D * Float64(M * Float64(0.5 / d))))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.125 * Float64(D / Float64(Float64(d * d) * l))) * h) * M) * D) * M)) * sqrt(d)) * sqrt(d)) / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))))) <= 2e+282)
		tmp = (1.0 - ((((0.25 * (M / d)) * D) / (l / h)) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * sqrt(d)) * sqrt(d)) / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+282], N[(N[(1.0 - N[(N[(N[(N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 - N[(N[(N[(N[(N[(0.125 * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 92.3

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

   6. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 92.3

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

   8. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-/r* N/A

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

      14. associate-/r/ N/A

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

      15. clear-num N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 91.8%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 46.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(1 - \left(\left(\left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot 0.125\right) \cdot h\right) \cdot M\right) \cdot D\right) \cdot M\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell}}}}{\sqrt{h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(1 - \frac{\left(0.25 \cdot \frac{M}{d}\right) \cdot D}{\frac{\ell}{h}} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 - \left(\left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot D\right) \cdot M\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(1 - \frac{\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 - \left(\left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot D\right) \cdot M\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
      2e+282)
   (*
    (- 1.0 (* (/ (* (* 0.25 D) (* (/ M d) h)) l) (* D (* M (/ 0.5 d)))))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (/
    (/
     (*
      (*
       (- 1.0 (* (* (* (* (* 0.125 (/ D (* (* d d) l))) h) M) D) M))
       (sqrt d))
      (sqrt d))
     (sqrt l))
    (sqrt h))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282) {
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * sqrt(d)) * sqrt(d)) / sqrt(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 / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))) <= 2d+282) then
        tmp = (1.0d0 - ((((0.25d0 * d_1) * ((m / d) * h)) / l) * (d_1 * (m * (0.5d0 / d))))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = ((((1.0d0 - (((((0.125d0 * (d_1 / ((d * d) * l))) * h) * m) * d_1) * m)) * sqrt(d)) * sqrt(d)) / sqrt(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 (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282) {
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * Math.sqrt(d)) * Math.sqrt(d)) / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282:
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * math.sqrt(d)) * math.sqrt(d)) / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0))))) <= 2e+282)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) / l) * Float64(D * Float64(M * Float64(0.5 / d))))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.125 * Float64(D / Float64(Float64(d * d) * l))) * h) * M) * D) * M)) * sqrt(d)) * sqrt(d)) / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))))) <= 2e+282)
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = ((((1.0 - (((((0.125 * (D / ((d * d) * l))) * h) * M) * D) * M)) * sqrt(d)) * sqrt(d)) / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+282], N[(N[(1.0 - N[(N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 - N[(N[(N[(N[(N[(0.125 * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 92.3

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

   6. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 92.3

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

   8. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-/r* N/A

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

      11. associate-/r/ N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

   10. Applied rewrites 89.6%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 46.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(1 - \left(\left(\left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot 0.125\right) \cdot h\right) \cdot M\right) \cdot D\right) \cdot M\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell}}}}{\sqrt{h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(1 - \frac{\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 - \left(\left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot D\right) \cdot M\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(1 - \left(\frac{\left(M \cdot 0.5\right) \cdot D}{\ell \cdot d} \cdot h\right) \cdot \left(\left(0.25 \cdot \frac{M}{d}\right) \cdot D\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
      2e+282)
   (*
    (- 1.0 (* (* (/ (* (* M 0.5) D) (* l d)) h) (* (* 0.25 (/ M d)) D)))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282) {
		tmp = (1.0 - (((((M * 0.5) * D) / (l * d)) * h) * ((0.25 * (M / d)) * D))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = d / (sqrt(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 / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))) <= 2d+282) then
        tmp = (1.0d0 - (((((m * 0.5d0) * d_1) / (l * d)) * h) * ((0.25d0 * (m / d)) * d_1))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = d / (sqrt(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 (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282) {
		tmp = (1.0 - (((((M * 0.5) * D) / (l * d)) * h) * ((0.25 * (M / d)) * D))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))) <= 2e+282:
		tmp = (1.0 - (((((M * 0.5) * D) / (l * d)) * h) * ((0.25 * (M / d)) * D))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0))))) <= 2e+282)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * 0.5) * D) / Float64(l * d)) * h) * Float64(Float64(0.25 * Float64(M / d)) * D))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))))) <= 2e+282)
		tmp = (1.0 - (((((M * 0.5) * D) / (l * d)) * h) * ((0.25 * (M / d)) * D))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+282], N[(N[(1.0 - N[(N[(N[(N[(N[(M * 0.5), $MachinePrecision] * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(0.25 * N[(M / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000007e282

   1. Initial program 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 92.3

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

   6. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 92.3

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

   8. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. /-rgt-identity N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 84.9%

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

   if 2.00000000000000007e282 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 22.8%

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

   3. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\left(1 - \left(\frac{\left(M \cdot 0.5\right) \cdot D}{\ell \cdot d} \cdot h\right) \cdot \left(\left(0.25 \cdot \frac{M}{d}\right) \cdot D\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{0.5}{d}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot t\_0}{\ell}\right) \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot t\_0\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ 0.5 d))) (t_1 (sqrt (- d))))
   (if (<= h -2.25e-61)
     (*
      (- 1.0 (* (/ (* (/ M d) (* (* D 0.5) 0.5)) (/ 1.0 h)) (/ (* D t_0) l)))
      (* (/ t_1 (sqrt (- l))) (sqrt (/ d h))))
     (if (<= h -1e-310)
       (*
        (* (/ t_1 (sqrt (- h))) (pow (/ d l) (/ 1.0 2.0)))
        (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
       (*
        (- 1.0 (* (/ D l) (* (* (* 0.25 D) (* (/ M d) h)) t_0)))
        (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (0.5 / d);
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -2.25e-61) {
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / sqrt(-l)) * sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = ((t_1 / sqrt(-h)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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) :: tmp
    t_0 = m * (0.5d0 / d)
    t_1 = sqrt(-d)
    if (h <= (-2.25d-61)) then
        tmp = (1.0d0 - ((((m / d) * ((d_1 * 0.5d0) * 0.5d0)) / (1.0d0 / h)) * ((d_1 * t_0) / l))) * ((t_1 / sqrt(-l)) * sqrt((d / h)))
    else if (h <= (-1d-310)) then
        tmp = ((t_1 / sqrt(-h)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    else
        tmp = (1.0d0 - ((d_1 / l) * (((0.25d0 * d_1) * ((m / d) * h)) * t_0))) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = M * (0.5 / d);
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (h <= -2.25e-61) {
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / Math.sqrt(-l)) * Math.sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = ((t_1 / Math.sqrt(-h)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (0.5 / d)
	t_1 = math.sqrt(-d)
	tmp = 0
	if h <= -2.25e-61:
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / math.sqrt(-l)) * math.sqrt((d / h)))
	elif h <= -1e-310:
		tmp = ((t_1 / math.sqrt(-h)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	else:
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(0.5 / d))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2.25e-61)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / Float64(1.0 / h)) * Float64(Float64(D * t_0) / l))) * Float64(Float64(t_1 / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(D / l) * Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) * t_0))) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (0.5 / d);
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (h <= -2.25e-61)
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((t_1 / sqrt(-l)) * sqrt((d / h)));
	elseif (h <= -1e-310)
		tmp = ((t_1 / sqrt(-h)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	else
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2.25e-61], N[(N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(D / l), $MachinePrecision] * N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.25e-61

   1. Initial program 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 80.2

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

   8. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      9. lower-neg.f64 91.5

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   10. Applied rewrites 91.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if -2.25e-61 < h < -9.999999999999969e-311

   1. Initial program 73.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-neg.f64 91.3

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 91.3%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -9.999999999999969e-311 < h

   1. Initial program 60.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 71.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 74.2

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 74.2%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 83.5%

       \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \frac{D}{\ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{0.5}{d}\\ t_1 := 1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot t\_0}{\ell}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -5.6 \cdot 10^{-131}:\\ \;\;\;\;t\_1 \cdot \left(\frac{t\_2}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{-1}{h}} \cdot t\_2\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot t\_0\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ 0.5 d)))
        (t_1
         (-
          1.0
          (* (/ (* (/ M d) (* (* D 0.5) 0.5)) (/ 1.0 h)) (/ (* D t_0) l))))
        (t_2 (sqrt (- d))))
   (if (<= h -5.6e-131)
     (* t_1 (* (/ t_2 (sqrt (- l))) (sqrt (/ d h))))
     (if (<= h -1e-310)
       (* (* (sqrt (/ d l)) (* (sqrt (/ -1.0 h)) t_2)) t_1)
       (*
        (- 1.0 (* (/ D l) (* (* (* 0.25 D) (* (/ M d) h)) t_0)))
        (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (0.5 / d);
	double t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l));
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -5.6e-131) {
		tmp = t_1 * ((t_2 / sqrt(-l)) * sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = (sqrt((d / l)) * (sqrt((-1.0 / h)) * t_2)) * t_1;
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = m * (0.5d0 / d)
    t_1 = 1.0d0 - ((((m / d) * ((d_1 * 0.5d0) * 0.5d0)) / (1.0d0 / h)) * ((d_1 * t_0) / l))
    t_2 = sqrt(-d)
    if (h <= (-5.6d-131)) then
        tmp = t_1 * ((t_2 / sqrt(-l)) * sqrt((d / h)))
    else if (h <= (-1d-310)) then
        tmp = (sqrt((d / l)) * (sqrt(((-1.0d0) / h)) * t_2)) * t_1
    else
        tmp = (1.0d0 - ((d_1 / l) * (((0.25d0 * d_1) * ((m / d) * h)) * t_0))) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = M * (0.5 / d);
	double t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (h <= -5.6e-131) {
		tmp = t_1 * ((t_2 / Math.sqrt(-l)) * Math.sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = (Math.sqrt((d / l)) * (Math.sqrt((-1.0 / h)) * t_2)) * t_1;
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (0.5 / d)
	t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))
	t_2 = math.sqrt(-d)
	tmp = 0
	if h <= -5.6e-131:
		tmp = t_1 * ((t_2 / math.sqrt(-l)) * math.sqrt((d / h)))
	elif h <= -1e-310:
		tmp = (math.sqrt((d / l)) * (math.sqrt((-1.0 / h)) * t_2)) * t_1
	else:
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(0.5 / d))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / Float64(1.0 / h)) * Float64(Float64(D * t_0) / l)))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -5.6e-131)
		tmp = Float64(t_1 * Float64(Float64(t_2 / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-1.0 / h)) * t_2)) * t_1);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(D / l) * Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) * t_0))) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (0.5 / d);
	t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (h <= -5.6e-131)
		tmp = t_1 * ((t_2 / sqrt(-l)) * sqrt((d / h)));
	elseif (h <= -1e-310)
		tmp = (sqrt((d / l)) * (sqrt((-1.0 / h)) * t_2)) * t_1;
	else
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -5.6e-131], N[(t$95$1 * N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(1.0 - N[(N[(D / l), $MachinePrecision] * N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.5999999999999999e-131

   1. Initial program 77.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 79.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 79.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 79.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 79.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 79.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 79.2

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 79.2%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      9. lower-neg.f64 90.0

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   10. Applied rewrites 90.0%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if -5.5999999999999999e-131 < h < -9.999999999999969e-311

   1. Initial program 72.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 72.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 72.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 72.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 72.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 72.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      7. div-inv N/A

         \[\leadsto \left(\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \left(\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{h}\right)\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{h}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-prod N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\mathsf{neg}\left(\frac{1}{h}\right)}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\mathsf{neg}\left(\frac{1}{h}\right)}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\mathsf{neg}\left(\frac{1}{h}\right)}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\mathsf{neg}\left(\frac{1}{h}\right)}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\frac{1}{h}\right)}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\mathsf{neg}\left(\color{blue}{\frac{1}{h}}\right)}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{h}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{\color{blue}{-1}}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      18. lower-/.f64 94.2

          \[\leadsto \left(\left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{h}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 94.2%

      \[\leadsto \left(\color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{h}}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if -9.999999999999969e-311 < h

   1. Initial program 60.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 71.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 74.2

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 74.2%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 83.5%

       \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \frac{D}{\ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5.6 \cdot 10^{-131}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{-1}{h}} \cdot \sqrt{-d}\right)\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{0.5}{d}\\ t_1 := 1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot t\_0}{\ell}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;t\_1 \cdot \left(\frac{t\_2}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{t\_2}{\sqrt{-h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot t\_0\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ 0.5 d)))
        (t_1
         (-
          1.0
          (* (/ (* (/ M d) (* (* D 0.5) 0.5)) (/ 1.0 h)) (/ (* D t_0) l))))
        (t_2 (sqrt (- d))))
   (if (<= h -2.25e-61)
     (* t_1 (* (/ t_2 (sqrt (- l))) (sqrt (/ d h))))
     (if (<= h -1e-310)
       (* (* (sqrt (/ d l)) (/ t_2 (sqrt (- h)))) t_1)
       (*
        (- 1.0 (* (/ D l) (* (* (* 0.25 D) (* (/ M d) h)) t_0)))
        (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (0.5 / d);
	double t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l));
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -2.25e-61) {
		tmp = t_1 * ((t_2 / sqrt(-l)) * sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = (sqrt((d / l)) * (t_2 / sqrt(-h))) * t_1;
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = m * (0.5d0 / d)
    t_1 = 1.0d0 - ((((m / d) * ((d_1 * 0.5d0) * 0.5d0)) / (1.0d0 / h)) * ((d_1 * t_0) / l))
    t_2 = sqrt(-d)
    if (h <= (-2.25d-61)) then
        tmp = t_1 * ((t_2 / sqrt(-l)) * sqrt((d / h)))
    else if (h <= (-1d-310)) then
        tmp = (sqrt((d / l)) * (t_2 / sqrt(-h))) * t_1
    else
        tmp = (1.0d0 - ((d_1 / l) * (((0.25d0 * d_1) * ((m / d) * h)) * t_0))) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = M * (0.5 / d);
	double t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (h <= -2.25e-61) {
		tmp = t_1 * ((t_2 / Math.sqrt(-l)) * Math.sqrt((d / h)));
	} else if (h <= -1e-310) {
		tmp = (Math.sqrt((d / l)) * (t_2 / Math.sqrt(-h))) * t_1;
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (0.5 / d)
	t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))
	t_2 = math.sqrt(-d)
	tmp = 0
	if h <= -2.25e-61:
		tmp = t_1 * ((t_2 / math.sqrt(-l)) * math.sqrt((d / h)))
	elif h <= -1e-310:
		tmp = (math.sqrt((d / l)) * (t_2 / math.sqrt(-h))) * t_1
	else:
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(0.5 / d))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / Float64(1.0 / h)) * Float64(Float64(D * t_0) / l)))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2.25e-61)
		tmp = Float64(t_1 * Float64(Float64(t_2 / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(t_2 / sqrt(Float64(-h)))) * t_1);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(D / l) * Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) * t_0))) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (0.5 / d);
	t_1 = 1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (h <= -2.25e-61)
		tmp = t_1 * ((t_2 / sqrt(-l)) * sqrt((d / h)));
	elseif (h <= -1e-310)
		tmp = (sqrt((d / l)) * (t_2 / sqrt(-h))) * t_1;
	else
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2.25e-61], N[(t$95$1 * N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(1.0 - N[(N[(D / l), $MachinePrecision] * N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.25e-61

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 80.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 80.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 80.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 80.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 80.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 80.2

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 80.2%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      9. lower-neg.f64 91.5

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   10. Applied rewrites 91.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if -2.25e-61 < h < -9.999999999999969e-311

   1. Initial program 73.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 72.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 72.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 72.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 72.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 72.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      12. lower-neg.f64 90.6

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 90.6%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if -9.999999999999969e-311 < h

   1. Initial program 60.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 71.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 74.2

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 74.2%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 83.5%

       \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \frac{D}{\ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{0.5}{d}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot t\_0}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot t\_0\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ 0.5 d))))
   (if (<= h -1e-310)
     (*
      (- 1.0 (* (/ (* (/ M d) (* (* D 0.5) 0.5)) (/ 1.0 h)) (/ (* D t_0) l)))
      (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h))))
     (*
      (- 1.0 (* (/ D l) (* (* (* 0.25 D) (* (/ M d) h)) t_0)))
      (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (0.5 / d);
	double tmp;
	if (h <= -1e-310) {
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((sqrt(-d) / sqrt(-l)) * sqrt((d / h)));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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) :: tmp
    t_0 = m * (0.5d0 / d)
    if (h <= (-1d-310)) then
        tmp = (1.0d0 - ((((m / d) * ((d_1 * 0.5d0) * 0.5d0)) / (1.0d0 / h)) * ((d_1 * t_0) / l))) * ((sqrt(-d) / sqrt(-l)) * sqrt((d / h)))
    else
        tmp = (1.0d0 - ((d_1 / l) * (((0.25d0 * d_1) * ((m / d) * h)) * t_0))) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = M * (0.5 / d);
	double tmp;
	if (h <= -1e-310) {
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h)));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (0.5 / d)
	tmp = 0
	if h <= -1e-310:
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h)))
	else:
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(0.5 / d))
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / Float64(1.0 / h)) * Float64(Float64(D * t_0) / l))) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(D / l) * Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) * t_0))) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (0.5 / d);
	tmp = 0.0;
	if (h <= -1e-310)
		tmp = (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l))) * ((sqrt(-d) / sqrt(-l)) * sqrt((d / h)));
	else
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1e-310], N[(N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(D / l), $MachinePrecision] * N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 75.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 77.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 77.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 77.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      9. lower-neg.f64 86.5

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   10. Applied rewrites 86.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if -9.999999999999969e-311 < h

   1. Initial program 60.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 71.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 71.9

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 74.2

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 74.2%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 83.5%

       \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \frac{D}{\ell}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{0.5}{d}\\ \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-305}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot t\_0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot t\_0\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ 0.5 d))))
   (if (<= l 3.5e-305)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (/ (* (/ M d) (* (* D 0.5) 0.5)) (/ 1.0 h)) (/ (* D t_0) l))))
     (*
      (- 1.0 (* (/ D l) (* (* (* 0.25 D) (* (/ M d) h)) t_0)))
      (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (0.5 / d);
	double tmp;
	if (l <= 3.5e-305) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l)));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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) :: tmp
    t_0 = m * (0.5d0 / d)
    if (l <= 3.5d-305) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((m / d) * ((d_1 * 0.5d0) * 0.5d0)) / (1.0d0 / h)) * ((d_1 * t_0) / l)))
    else
        tmp = (1.0d0 - ((d_1 / l) * (((0.25d0 * d_1) * ((m / d) * h)) * t_0))) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = M * (0.5 / d);
	double tmp;
	if (l <= 3.5e-305) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l)));
	} else {
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (0.5 / d)
	tmp = 0
	if l <= 3.5e-305:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l)))
	else:
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(0.5 / d))
	tmp = 0.0
	if (l <= 3.5e-305)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / Float64(1.0 / h)) * Float64(Float64(D * t_0) / l))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(D / l) * Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) * t_0))) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (0.5 / d);
	tmp = 0.0;
	if (l <= 3.5e-305)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * t_0) / l)));
	else
		tmp = (1.0 - ((D / l) * (((0.25 * D) * ((M / d) * h)) * t_0))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3.5e-305], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(D / l), $MachinePrecision] * N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.4999999999999998e-305

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 77.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 77.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.6

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 77.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if 3.4999999999999998e-305 < l

   1. Initial program 60.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 72.2

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 72.2%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 72.2

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 74.5

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 74.5%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.2%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 83.3%

       \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \frac{D}{\ell}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-305}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{D}{\ell} \cdot \left(\left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{-177}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(M \cdot 0.5\right) \cdot D}{\ell \cdot d} \cdot \left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.45e-177)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (-
     1.0
     (*
      (/ (* (/ M d) (* (* D 0.5) 0.5)) (/ 1.0 h))
      (/ (* D (* M (/ 0.5 d))) l))))
   (*
    (- 1.0 (* (/ (* (* M 0.5) D) (* l d)) (* (* 0.25 D) (* (/ M d) h))))
    (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.45e-177) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * (M * (0.5 / d))) / l)));
	} else {
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * ((0.25 * D) * ((M / d) * h)))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 <= 2.45d-177) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((m / d) * ((d_1 * 0.5d0) * 0.5d0)) / (1.0d0 / h)) * ((d_1 * (m * (0.5d0 / d))) / l)))
    else
        tmp = (1.0d0 - ((((m * 0.5d0) * d_1) / (l * d)) * ((0.25d0 * d_1) * ((m / d) * h)))) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 <= 2.45e-177) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * (M * (0.5 / d))) / l)));
	} else {
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * ((0.25 * D) * ((M / d) * h)))) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.45e-177:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * (M * (0.5 / d))) / l)))
	else:
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * ((0.25 * D) * ((M / d) * h)))) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.45e-177)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / Float64(1.0 / h)) * Float64(Float64(D * Float64(M * Float64(0.5 / d))) / l))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M * 0.5) * D) / Float64(l * d)) * Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)))) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2.45e-177)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((M / d) * ((D * 0.5) * 0.5)) / (1.0 / h)) * ((D * (M * (0.5 / d))) / l)));
	else
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * ((0.25 * D) * ((M / d) * h)))) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.45e-177], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(M * 0.5), $MachinePrecision] * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.44999999999999994e-177

   1. Initial program 72.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 77.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 77.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 77.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   if 2.44999999999999994e-177 < l

   1. Initial program 59.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 73.0

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 73.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 73.0

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 75.3

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 75.3%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 76.2%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 84.0%

       \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot 0.5\right) \cdot D}{d \cdot \ell} \cdot \left(\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{-177}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\frac{1}{h}} \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(M \cdot 0.5\right) \cdot D}{\ell \cdot d} \cdot \left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\\ \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 - \frac{t\_0}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(M \cdot 0.5\right) \cdot D}{\ell \cdot d} \cdot t\_0\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* 0.25 D) (* (/ M d) h))))
   (if (<= l 1.35e-272)
     (*
      (- 1.0 (* (/ t_0 l) (* D (* M (/ 0.5 d)))))
      (* (sqrt (/ d l)) (sqrt (/ d h))))
     (*
      (- 1.0 (* (/ (* (* M 0.5) D) (* l d)) t_0))
      (* (* (/ 1.0 (sqrt l)) (sqrt d)) (/ (sqrt d) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.25 * D) * ((M / d) * h);
	double tmp;
	if (l <= 1.35e-272) {
		tmp = (1.0 - ((t_0 / l) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * t_0)) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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) :: tmp
    t_0 = (0.25d0 * d_1) * ((m / d) * h)
    if (l <= 1.35d-272) then
        tmp = (1.0d0 - ((t_0 / l) * (d_1 * (m * (0.5d0 / d))))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = (1.0d0 - ((((m * 0.5d0) * d_1) / (l * d)) * t_0)) * (((1.0d0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / 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 = (0.25 * D) * ((M / d) * h);
	double tmp;
	if (l <= 1.35e-272) {
		tmp = (1.0 - ((t_0 / l) * (D * (M * (0.5 / d))))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * t_0)) * (((1.0 / Math.sqrt(l)) * Math.sqrt(d)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (0.25 * D) * ((M / d) * h)
	tmp = 0
	if l <= 1.35e-272:
		tmp = (1.0 - ((t_0 / l) * (D * (M * (0.5 / d))))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * t_0)) * (((1.0 / math.sqrt(l)) * math.sqrt(d)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h))
	tmp = 0.0
	if (l <= 1.35e-272)
		tmp = Float64(Float64(1.0 - Float64(Float64(t_0 / l) * Float64(D * Float64(M * Float64(0.5 / d))))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M * 0.5) * D) / Float64(l * d)) * t_0)) * Float64(Float64(Float64(1.0 / sqrt(l)) * sqrt(d)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.25 * D) * ((M / d) * h);
	tmp = 0.0;
	if (l <= 1.35e-272)
		tmp = (1.0 - ((t_0 / l) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = (1.0 - ((((M * 0.5) * D) / (l * d)) * t_0)) * (((1.0 / sqrt(l)) * sqrt(d)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.35e-272], N[(N[(1.0 - N[(N[(t$95$0 / l), $MachinePrecision] * N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(M * 0.5), $MachinePrecision] * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.34999999999999996e-272

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 77.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 77.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 77.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 77.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 77.9%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell}} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D\right) \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \left(M \cdot D\right)\right)} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2 \cdot d}} \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      11. associate-/r/ N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{\frac{2 \cdot d}{M \cdot D}}} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}}\right) \]

   10. Applied rewrites 75.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \frac{\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)}{\ell}}\right) \]

   if 1.34999999999999996e-272 < l

   1. Initial program 60.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 73.1

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 73.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 73.1

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 74.9

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 74.9%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 83.3%

       \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot 0.5\right) \cdot D}{d \cdot \ell} \cdot \left(\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 - \frac{\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(M \cdot 0.5\right) \cdot D}{\ell \cdot d} \cdot \left(\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 63.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \frac{D}{\left(d \cdot d\right) \cdot \ell}\\ t_3 := 1 - \left(\left(\left(0.125 \cdot t\_2\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\\ t_4 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-150}:\\ \;\;\;\;\left(t\_3 \cdot t\_0\right) \cdot t\_1\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{t\_4}\right)\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(-0.125 \cdot h\right) \cdot \left(\left(t\_2 \cdot M\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{d}{t\_4}\\ \mathbf{elif}\;d \leq 98000:\\ \;\;\;\;\left(1 - \frac{\left(\frac{M \cdot M}{d \cdot d} \cdot \left(\left(D \cdot D\right) \cdot 0.125\right)\right) \cdot h}{\ell}\right) \cdot \left(t\_1 \cdot t\_0\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2 (/ D (* (* d d) l)))
        (t_3 (- 1.0 (* (* (* (* 0.125 t_2) h) M) (* D M))))
        (t_4 (sqrt (* l h))))
   (if (<= d -1.02e+155)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= d -4.6e-150)
       (* (* t_3 t_0) t_1)
       (if (<= d -2.8e-303)
         (fma
          (* (sqrt (/ h (* (* l l) l))) D)
          (* (- D) (* (* -0.125 (/ M d)) M))
          (/ (- d) t_4))
         (if (<= d 4.1e-162)
           (* (* (* -0.125 h) (* (* t_2 M) (* D M))) (/ d t_4))
           (if (<= d 98000.0)
             (*
              (- 1.0 (/ (* (* (/ (* M M) (* d d)) (* (* D D) 0.125)) h) l))
              (* t_1 t_0))
             (if (<= d 9e+110)
               (* (sqrt (/ (* d d) (* l h))) t_3)
               (/ d (* (sqrt l) (sqrt h)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = D / ((d * d) * l);
	double t_3 = 1.0 - ((((0.125 * t_2) * h) * M) * (D * M));
	double t_4 = sqrt((l * h));
	double tmp;
	if (d <= -1.02e+155) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (d <= -4.6e-150) {
		tmp = (t_3 * t_0) * t_1;
	} else if (d <= -2.8e-303) {
		tmp = fma((sqrt((h / ((l * l) * l))) * D), (-D * ((-0.125 * (M / d)) * M)), (-d / t_4));
	} else if (d <= 4.1e-162) {
		tmp = ((-0.125 * h) * ((t_2 * M) * (D * M))) * (d / t_4);
	} else if (d <= 98000.0) {
		tmp = (1.0 - (((((M * M) / (d * d)) * ((D * D) * 0.125)) * h) / l)) * (t_1 * t_0);
	} else if (d <= 9e+110) {
		tmp = sqrt(((d * d) / (l * h))) * t_3;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(D / Float64(Float64(d * d) * l))
	t_3 = Float64(1.0 - Float64(Float64(Float64(Float64(0.125 * t_2) * h) * M) * Float64(D * M)))
	t_4 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -1.02e+155)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (d <= -4.6e-150)
		tmp = Float64(Float64(t_3 * t_0) * t_1);
	elseif (d <= -2.8e-303)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * D), Float64(Float64(-D) * Float64(Float64(-0.125 * Float64(M / d)) * M)), Float64(Float64(-d) / t_4));
	elseif (d <= 4.1e-162)
		tmp = Float64(Float64(Float64(-0.125 * h) * Float64(Float64(t_2 * M) * Float64(D * M))) * Float64(d / t_4));
	elseif (d <= 98000.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * M) / Float64(d * d)) * Float64(Float64(D * D) * 0.125)) * h) / l)) * Float64(t_1 * t_0));
	elseif (d <= 9e+110)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(l * h))) * t_3);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[(N[(N[(0.125 * t$95$2), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.02e+155], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, -4.6e-150], N[(N[(t$95$3 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, -2.8e-303], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D), $MachinePrecision] * N[((-D) * N[(N[(-0.125 * N[(M / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] + N[((-d) / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.1e-162], N[(N[(N[(-0.125 * h), $MachinePrecision] * N[(N[(t$95$2 * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 98000.0], N[(N[(1.0 - N[(N[(N[(N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9e+110], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if d < -1.02e155

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 38.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 56.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.02e155 < d < -4.60000000000000006e-150

   1. Initial program 86.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 77.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -4.60000000000000006e-150 < d < -2.8e-303

   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. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 28.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)}{\sqrt{h}}} \]

   8. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \frac{-1}{8}, \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot D\right)\right) \cdot \left(\left(-M\right) \cdot \frac{M}{d}\right), -0.125, \left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 78.1%

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \color{blue}{D \cdot \left(\left(-M\right) \cdot \left(\frac{M}{d} \cdot -0.125\right)\right)}, \frac{-d}{\sqrt{\ell \cdot h}}\right) \]

   if -2.8e-303 < d < 4.10000000000000019e-162

   1. Initial program 34.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 51.3

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 51.3%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in h around inf

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{-1}{8} \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

      2. associate-*l/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} \cdot h\right)}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(h \cdot \frac{-1}{8}\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(h \cdot \frac{-1}{8}\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(h \cdot \frac{-1}{8}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2} \cdot \ell}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \frac{{M}^{2} \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{\left({M}^{2} \cdot D\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot D\right) \cdot \frac{D}{{d}^{2} \cdot \ell}\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot D\right) \cdot \frac{D}{{d}^{2} \cdot \ell}\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \left(\color{blue}{\left({M}^{2} \cdot D\right)} \cdot \frac{D}{{d}^{2} \cdot \ell}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot D\right) \cdot \frac{D}{{d}^{2} \cdot \ell}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot D\right) \cdot \frac{D}{{d}^{2} \cdot \ell}\right)\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \color{blue}{\frac{D}{{d}^{2} \cdot \ell}}\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right)\right) \]

      19. lower-*.f64 30.7

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(h \cdot -0.125\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right)\right) \]

   7. Applied rewrites 30.7%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(h \cdot -0.125\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right)} \]

   9. Applied rewrites 34.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(\left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \left(-0.125 \cdot h\right)\right)} \]

   if 4.10000000000000019e-162 < d < 98000

   1. Initial program 74.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 88.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 88.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 88.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 88.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 88.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 88.5

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 88.5%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   10. Applied rewrites 80.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d \cdot d}\right) \cdot h}{\ell}}\right) \]

   if 98000 < d < 9.0000000000000005e110

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 75.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

   if 9.0000000000000005e110 < d

   1. Initial program 62.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 45.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 75.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 78.0%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{\sqrt{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(-0.125 \cdot h\right) \cdot \left(\left(\frac{D}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 98000:\\ \;\;\;\;\left(1 - \frac{\left(\frac{M \cdot M}{d \cdot d} \cdot \left(\left(D \cdot D\right) \cdot 0.125\right)\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}} \cdot \left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \frac{M}{d} \cdot D\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\left(t\_1 \cdot t\_1\right) \cdot 0.25\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))) (t_1 (* (/ M d) D)))
   (if (<= d -1.02e+155)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= d -4.6e-150)
       (*
        (*
         (- 1.0 (* (* (* (* 0.125 (/ D (* (* d d) l))) h) M) (* D M)))
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (if (<= d -2.8e-303)
         (fma
          (* (sqrt (/ h (* (* l l) l))) D)
          (* (- D) (* (* -0.125 (/ M d)) M))
          (/ (- d) t_0))
         (*
          (- 1.0 (* (* (* (* t_1 t_1) 0.25) (/ 1.0 2.0)) (/ h l)))
          (/ d t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = (M / d) * D;
	double tmp;
	if (d <= -1.02e+155) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (d <= -4.6e-150) {
		tmp = ((1.0 - ((((0.125 * (D / ((d * d) * l))) * h) * M) * (D * M))) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -2.8e-303) {
		tmp = fma((sqrt((h / ((l * l) * l))) * D), (-D * ((-0.125 * (M / d)) * M)), (-d / t_0));
	} else {
		tmp = (1.0 - ((((t_1 * t_1) * 0.25) * (1.0 / 2.0)) * (h / l))) * (d / t_0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64(M / d) * D)
	tmp = 0.0
	if (d <= -1.02e+155)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (d <= -4.6e-150)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.125 * Float64(D / Float64(Float64(d * d) * l))) * h) * M) * Float64(D * M))) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -2.8e-303)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * D), Float64(Float64(-D) * Float64(Float64(-0.125 * Float64(M / d)) * M)), Float64(Float64(-d) / t_0));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) * 0.25) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64(d / t_0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, If[LessEqual[d, -1.02e+155], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, -4.6e-150], N[(N[(N[(1.0 - N[(N[(N[(N[(0.125 * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.8e-303], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D), $MachinePrecision] * N[((-D) * N[(N[(-0.125 * N[(M / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] + N[((-d) / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.25), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.02e155

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 38.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 56.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.02e155 < d < -4.60000000000000006e-150

   1. Initial program 86.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 77.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -4.60000000000000006e-150 < d < -2.8e-303

   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. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 28.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)}{\sqrt{h}}} \]

   8. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \frac{-1}{8}, \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot D\right)\right) \cdot \left(\left(-M\right) \cdot \frac{M}{d}\right), -0.125, \left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 78.1%

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \color{blue}{D \cdot \left(\left(-M\right) \cdot \left(\frac{M}{d} \cdot -0.125\right)\right)}, \frac{-d}{\sqrt{\ell \cdot h}}\right) \]

   if -2.8e-303 < d

   1. Initial program 60.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 71.4

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 71.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 71.4

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 73.6

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.0%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{1 \cdot \sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(1 \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d}} \cdot \left(1 \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 67.1

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 67.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{\sqrt{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot 0.25\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-150}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{\sqrt{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 98000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot d} \cdot \left(D \cdot D\right)\right) \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* (* (* 0.125 (/ D (* (* d d) l))) h) M) (* D M)))))
   (if (<= d -1.02e+155)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= d -4.6e-150)
       (* (* t_0 (sqrt (/ d h))) (sqrt (/ d l)))
       (if (<= d -2.8e-303)
         (fma
          (* (sqrt (/ h (* (* l l) l))) D)
          (* (- D) (* (* -0.125 (/ M d)) M))
          (/ (- d) (sqrt (* l h))))
         (if (<= d 98000.0)
           (/
            (*
             (fma (* (* (/ (* M M) (* d d)) (* D D)) -0.125) (/ h l) 1.0)
             (/ d (sqrt l)))
            (sqrt h))
           (if (<= d 9e+110)
             (* (sqrt (/ (* d d) (* l h))) t_0)
             (/ d (* (sqrt l) (sqrt h))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - ((((0.125 * (D / ((d * d) * l))) * h) * M) * (D * M));
	double tmp;
	if (d <= -1.02e+155) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (d <= -4.6e-150) {
		tmp = (t_0 * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= -2.8e-303) {
		tmp = fma((sqrt((h / ((l * l) * l))) * D), (-D * ((-0.125 * (M / d)) * M)), (-d / sqrt((l * h))));
	} else if (d <= 98000.0) {
		tmp = (fma(((((M * M) / (d * d)) * (D * D)) * -0.125), (h / l), 1.0) * (d / sqrt(l))) / sqrt(h);
	} else if (d <= 9e+110) {
		tmp = sqrt(((d * d) / (l * h))) * t_0;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(Float64(0.125 * Float64(D / Float64(Float64(d * d) * l))) * h) * M) * Float64(D * M)))
	tmp = 0.0
	if (d <= -1.02e+155)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (d <= -4.6e-150)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= -2.8e-303)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * D), Float64(Float64(-D) * Float64(Float64(-0.125 * Float64(M / d)) * M)), Float64(Float64(-d) / sqrt(Float64(l * h))));
	elseif (d <= 98000.0)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(M * M) / Float64(d * d)) * Float64(D * D)) * -0.125), Float64(h / l), 1.0) * Float64(d / sqrt(l))) / sqrt(h));
	elseif (d <= 9e+110)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(l * h))) * t_0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[(0.125 * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.02e+155], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, -4.6e-150], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.8e-303], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D), $MachinePrecision] * N[((-D) * N[(N[(-0.125 * N[(M / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] + N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 98000.0], N[(N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9e+110], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -1.02e155

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 38.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 56.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.02e155 < d < -4.60000000000000006e-150

   1. Initial program 86.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 77.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -4.60000000000000006e-150 < d < -2.8e-303

   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. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 28.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)}{\sqrt{h}}} \]

   8. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \frac{-1}{8}, \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot D\right)\right) \cdot \left(\left(-M\right) \cdot \frac{M}{d}\right), -0.125, \left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 78.1%

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \color{blue}{D \cdot \left(\left(-M\right) \cdot \left(\frac{M}{d} \cdot -0.125\right)\right)}, \frac{-d}{\sqrt{\ell \cdot h}}\right) \]

   if -2.8e-303 < d < 98000

   1. Initial program 56.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 65.4

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 65.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 65.4

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 66.3

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 67.7%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 49.8%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot \left(\frac{M \cdot M}{d \cdot d} \cdot \left(D \cdot D\right)\right), \frac{h}{\ell}, 1\right)}{\sqrt{h}}} \]

   if 98000 < d < 9.0000000000000005e110

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 75.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

   if 9.0000000000000005e110 < d

   1. Initial program 62.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 45.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 75.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 75.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 78.0%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{\sqrt{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 98000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot d} \cdot \left(D \cdot D\right)\right) \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}} \cdot \left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 68.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ \mathbf{if}\;d \leq 1.2 \cdot 10^{+74}:\\ \;\;\;\;\left(1 - \frac{\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\left(t\_0 \cdot t\_0\right) \cdot 0.25\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ M d) D)))
   (if (<= d 1.2e+74)
     (*
      (- 1.0 (* (/ (* (* 0.25 D) (* (/ M d) h)) l) (* D (* M (/ 0.5 d)))))
      (* (sqrt (/ d l)) (sqrt (/ d h))))
     (*
      (- 1.0 (* (* (* (* t_0 t_0) 0.25) (/ 1.0 2.0)) (/ h l)))
      (/ d (sqrt (* l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M / d) * D;
	double tmp;
	if (d <= 1.2e+74) {
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = (1.0 - ((((t_0 * t_0) * 0.25) * (1.0 / 2.0)) * (h / l))) * (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) :: t_0
    real(8) :: tmp
    t_0 = (m / d) * d_1
    if (d <= 1.2d+74) then
        tmp = (1.0d0 - ((((0.25d0 * d_1) * ((m / d) * h)) / l) * (d_1 * (m * (0.5d0 / d))))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = (1.0d0 - ((((t_0 * t_0) * 0.25d0) * (1.0d0 / 2.0d0)) * (h / l))) * (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 t_0 = (M / d) * D;
	double tmp;
	if (d <= 1.2e+74) {
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = (1.0 - ((((t_0 * t_0) * 0.25) * (1.0 / 2.0)) * (h / l))) * (d / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M / d) * D
	tmp = 0
	if d <= 1.2e+74:
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = (1.0 - ((((t_0 * t_0) * 0.25) * (1.0 / 2.0)) * (h / l))) * (d / math.sqrt((l * h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / d) * D)
	tmp = 0.0
	if (d <= 1.2e+74)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D) * Float64(Float64(M / d) * h)) / l) * Float64(D * Float64(M * Float64(0.5 / d))))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.25) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M / d) * D;
	tmp = 0.0;
	if (d <= 1.2e+74)
		tmp = (1.0 - ((((0.25 * D) * ((M / d) * h)) / l) * (D * (M * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = (1.0 - ((((t_0 * t_0) * 0.25) * (1.0 / 2.0)) * (h / l))) * (d / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, If[LessEqual[d, 1.2e+74], N[(N[(1.0 - N[(N[(N[(N[(0.25 * D), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.25), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.20000000000000004e74

   1. Initial program 68.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 75.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 75.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 75.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 75.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      2. metadata-eval 75.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      5. lower-sqrt.f64 75.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 75.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{\ell}} \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D\right) \cdot \frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D\right)} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)} \cdot D\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \left(M \cdot D\right)\right)} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2 \cdot d}} \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      11. associate-/r/ N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{\frac{2 \cdot d}{M \cdot D}}} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}}{\frac{1}{h}}}{\ell}}\right) \]

   10. Applied rewrites 73.0%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \frac{\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right)}{\ell}}\right) \]

   if 1.20000000000000004e74 < d

   1. Initial program 64.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 81.8

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 81.8%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 81.8

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 86.6

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 86.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{1 \cdot \sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(1 \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d}} \cdot \left(1 \cdot \sqrt{d}\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 79.8

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 79.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.2 \cdot 10^{+74}:\\ \;\;\;\;\left(1 - \frac{\left(0.25 \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right)}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot 0.25\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 57.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -9.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)}{\sqrt{\frac{h}{\frac{d}{\ell} \cdot d}}}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{t\_0}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot d} \cdot \left(D \cdot D\right)\right) \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= d -9.4e+154)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (if (<= d -1.5e-127)
       (/
        (- 1.0 (* (* (* (* 0.125 (/ D (* (* d d) l))) h) M) (* D M)))
        (sqrt (/ h (* (/ d l) d))))
       (if (<= d -2.8e-303)
         (fma
          (* (sqrt (/ h (* (* l l) l))) D)
          (* (- D) (* (* -0.125 (/ M d)) M))
          (/ (- d) t_0))
         (if (<= d 3.8e-51)
           (*
            (fma (* (* (/ (* M M) (* d d)) (* D D)) -0.125) (/ h l) 1.0)
            (/ d t_0))
           (/ d (* (sqrt l) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -9.4e+154) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (d <= -1.5e-127) {
		tmp = (1.0 - ((((0.125 * (D / ((d * d) * l))) * h) * M) * (D * M))) / sqrt((h / ((d / l) * d)));
	} else if (d <= -2.8e-303) {
		tmp = fma((sqrt((h / ((l * l) * l))) * D), (-D * ((-0.125 * (M / d)) * M)), (-d / t_0));
	} else if (d <= 3.8e-51) {
		tmp = fma(((((M * M) / (d * d)) * (D * D)) * -0.125), (h / l), 1.0) * (d / t_0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -9.4e+154)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (d <= -1.5e-127)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.125 * Float64(D / Float64(Float64(d * d) * l))) * h) * M) * Float64(D * M))) / sqrt(Float64(h / Float64(Float64(d / l) * d))));
	elseif (d <= -2.8e-303)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * D), Float64(Float64(-D) * Float64(Float64(-0.125 * Float64(M / d)) * M)), Float64(Float64(-d) / t_0));
	elseif (d <= 3.8e-51)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M * M) / Float64(d * d)) * Float64(D * D)) * -0.125), Float64(h / l), 1.0) * Float64(d / t_0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -9.4e+154], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, -1.5e-127], N[(N[(1.0 - N[(N[(N[(N[(0.125 * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / N[(N[(d / l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.8e-303], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D), $MachinePrecision] * N[((-D) * N[(N[(-0.125 * N[(M / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] + N[((-d) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-51], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -9.39999999999999966e154

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 38.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 56.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -9.39999999999999966e154 < d < -1.50000000000000004e-127

   1. Initial program 88.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 79.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\frac{1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)}{\sqrt{\frac{h}{d \cdot \frac{d}{\ell}}}}} \]

   if -1.50000000000000004e-127 < d < -2.8e-303

   1. Initial program 63.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 37.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   7. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\left(1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)}{\sqrt{h}}} \]

   8. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} + \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \frac{-1}{8}, \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   10. Applied rewrites 50.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot D\right)\right) \cdot \left(\left(-M\right) \cdot \frac{M}{d}\right), -0.125, \left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.9%

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \color{blue}{D \cdot \left(\left(-M\right) \cdot \left(\frac{M}{d} \cdot -0.125\right)\right)}, \frac{-d}{\sqrt{\ell \cdot h}}\right) \]

   if -2.8e-303 < d < 3.80000000000000003e-51

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 63.9

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 63.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 63.9

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\frac{\ell}{d}}\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{1}{\ell} \cdot d\right)}}^{\frac{1}{2}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot {d}^{\frac{1}{2}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{d}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-sqrt.f64 64.9

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 64.9%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. div-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot {\left(\frac{1}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. swap-sqr N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{\frac{1}{2}}{d} \cdot M\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\frac{\frac{1}{2} \cdot M}{d}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(D \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(D \cdot \frac{1}{2}\right) \cdot \frac{M}{d}\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot D\right)} \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. associate-*l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. associate-*r* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2}}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{d}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 44.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125 \cdot \left(\frac{M \cdot M}{d \cdot d} \cdot \left(D \cdot D\right)\right), \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 3.80000000000000003e-51 < d

   1. Initial program 65.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 56.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 67.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 72.6%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{1 - \left(\left(\left(0.125 \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h\right) \cdot M\right) \cdot \left(D \cdot M\right)}{\sqrt{\frac{h}{\frac{d}{\ell} \cdot d}}}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot D, \left(-D\right) \cdot \left(\left(-0.125 \cdot \frac{M}{d}\right) \cdot M\right), \frac{-d}{\sqrt{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot d} \cdot \left(D \cdot D\right)\right) \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 47.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -250000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -250000000.0)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= h -1e-310)
     (* (sqrt (/ 1.0 (* l h))) (- d))
     (/ d (* (sqrt l) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -250000000.0) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (h <= -1e-310) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(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 (h <= (-250000000.0d0)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (h <= (-1d-310)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(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 (h <= -250000000.0) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (h <= -1e-310) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -250000000.0:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif h <= -1e-310:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -250000000.0)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (h <= -1e-310)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -250000000.0)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (h <= -1e-310)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -250000000.0], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.5e8

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 60.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 7.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 41.3%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   if -2.5e8 < h < -9.999999999999969e-311

   1. Initial program 73.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 48.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 53.2

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -9.999999999999969e-311 < h

   1. Initial program 60.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 48.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 48.7

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 48.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 53.6%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -250000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 45.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 5.5e-155)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 5.5e-155) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(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 <= 5.5d-155) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(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 <= 5.5e-155) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 5.5e-155:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 5.5e-155)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 5.5e-155)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 5.5e-155], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.50000000000000018e-155

   1. Initial program 68.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 48.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 36.3

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 36.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 5.50000000000000018e-155 < d

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 55.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 59.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 65.3%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 42.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 5.5e-155) (* (sqrt (/ 1.0 (* l h))) (- d)) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 5.5e-155) {
		tmp = sqrt((1.0 / (l * h))) * -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 (d <= 5.5d-155) then
        tmp = sqrt((1.0d0 / (l * h))) * -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 (d <= 5.5e-155) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 5.5e-155:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 5.5e-155)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-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 (d <= 5.5e-155)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 5.5e-155], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.50000000000000018e-155

   1. Initial program 68.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 48.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 36.3

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Applied rewrites 36.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 5.50000000000000018e-155 < d

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 55.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 59.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 34.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.3e-156) (sqrt (/ (* d d) (* l h))) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.3e-156) {
		tmp = sqrt(((d * d) / (l * h)));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.3d-156)) then
        tmp = sqrt(((d * d) / (l * h)))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.3e-156) {
		tmp = Math.sqrt(((d * d) / (l * h)));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.3e-156:
		tmp = math.sqrt(((d * d) / (l * h)))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.3e-156)
		tmp = sqrt(Float64(Float64(d * d) / Float64(l * h)));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.3e-156)
		tmp = sqrt(((d * d) / (l * h)));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.3e-156], N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.3e-156

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 3.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 24.1%

       \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

   if -1.3e-156 < l

   1. Initial program 64.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   5. Applied rewrites 46.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 43.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   2. unpow2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   3. associate-*l* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   4. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot D\right) \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   5. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{1}{8} \cdot D\right) \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

   6. associate-*l/ N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot D}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \frac{1}{8}}}{{d}^{2} \cdot \ell} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot {M}^{2}\right)} \cdot D\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot \color{blue}{\left(h \cdot \left({M}^{2} \cdot D\right)\right)}\right) \]

   10. associate-*r* N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot \frac{1}{8}}{{d}^{2} \cdot \ell} \cdot h\right) \cdot \left({M}^{2} \cdot D\right)}\right) \]

5. Applied rewrites 51.3%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{\ell \cdot \left(d \cdot d\right)} \cdot 0.125\right) \cdot h\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}\right) \]

6. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 30.5

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 30.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

10. Applied rewrites 30.6%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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)))))