Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.8% → 84.0%

Time: 21.8s

Alternatives: 24

Speedup: 2.9×

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: 67.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{-297}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot \frac{t\_1}{\sqrt{-\ell}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (*
           (/ (/ (* M D) (* d 2.0)) l)
           (/ (/ (* (* M D) 0.5) (* d 2.0)) (/ -1.0 h)))))
        (t_1 (sqrt (- d))))
   (if (<= d -1.25e-297)
     (* (* (/ t_1 (sqrt (- h))) (/ t_1 (sqrt (- l)))) t_0)
     (* t_0 (* (* (sqrt (/ 1.0 h)) (sqrt d)) (/ 1.0 (sqrt (/ l d))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)));
	double t_1 = sqrt(-d);
	double tmp;
	if (d <= -1.25e-297) {
		tmp = ((t_1 / sqrt(-h)) * (t_1 / sqrt(-l))) * t_0;
	} else {
		tmp = t_0 * ((sqrt((1.0 / h)) * sqrt(d)) * (1.0 / sqrt((l / d))));
	}
	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 = 1.0d0 + ((((m * d_1) / (d * 2.0d0)) / l) * ((((m * d_1) * 0.5d0) / (d * 2.0d0)) / ((-1.0d0) / h)))
    t_1 = sqrt(-d)
    if (d <= (-1.25d-297)) then
        tmp = ((t_1 / sqrt(-h)) * (t_1 / sqrt(-l))) * t_0
    else
        tmp = t_0 * ((sqrt((1.0d0 / h)) * sqrt(d)) * (1.0d0 / sqrt((l / d))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (d <= -1.25e-297) {
		tmp = ((t_1 / Math.sqrt(-h)) * (t_1 / Math.sqrt(-l))) * t_0;
	} else {
		tmp = t_0 * ((Math.sqrt((1.0 / h)) * Math.sqrt(d)) * (1.0 / Math.sqrt((l / d))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))
	t_1 = math.sqrt(-d)
	tmp = 0
	if d <= -1.25e-297:
		tmp = ((t_1 / math.sqrt(-h)) * (t_1 / math.sqrt(-l))) * t_0
	else:
		tmp = t_0 * ((math.sqrt((1.0 / h)) * math.sqrt(d)) * (1.0 / math.sqrt((l / d))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / l) * Float64(Float64(Float64(Float64(M * D) * 0.5) / Float64(d * 2.0)) / Float64(-1.0 / h))))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1.25e-297)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(t_1 / sqrt(Float64(-l)))) * t_0);
	else
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(1.0 / h)) * sqrt(d)) * Float64(1.0 / sqrt(Float64(l / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (d <= -1.25e-297)
		tmp = ((t_1 / sqrt(-h)) * (t_1 / sqrt(-l))) * t_0;
	else
		tmp = t_0 * ((sqrt((1.0 / h)) * sqrt(d)) * (1.0 / sqrt((l / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1.25e-297], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.25e-297

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied rewrites 73.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 82.8

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

   6. Applied rewrites 82.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 91.6

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

   8. Applied rewrites 91.6%

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

   if -1.25e-297 < d

   1. Initial program 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 75.6

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

   6. Applied rewrites 75.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. sqrt-pow1 N/A

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

      12. lower-*.f64 N/A

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

      13. sqrt-pow1 N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow1/2 N/A

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

      20. lower-sqrt.f64 84.2

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

   8. Applied rewrites 84.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-297}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq 1000000:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 1000000.0)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* h (/ (* (* M D) (* (* M D) 0.5)) (* (* d 2.0) (* l (* d 2.0))))))))
     (if (<= t_1 INFINITY)
       (/ (* d (sqrt (/ h l))) h)
       (* d (sqrt (sqrt (* t_0 t_0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= 1000000.0) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = d * sqrt(sqrt((t_0 * t_0)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= 1000000.0) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d * Math.sqrt(Math.sqrt((t_0 * t_0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 / (h * l)
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_1 <= 1000000.0:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))))
	elif t_1 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = d * math.sqrt(math.sqrt((t_0 * t_0)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	t_1 = 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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= 1000000.0)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * 0.5)) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0))))))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d * sqrt(sqrt(Float64(t_0 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 / (h * l);
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_1 <= 1000000.0)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))));
	elseif (t_1 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = d * sqrt(sqrt((t_0 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1000000.0], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $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)))) < 1e6

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied rewrites 87.5%

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

   6. Applied rewrites 73.2%

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

   if 1e6 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 81.2%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 28.6%

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

   9. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 86.9

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

   11. Applied rewrites 86.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 13.8

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

   8. Applied rewrites 13.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 13.8

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 1000000:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_1 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right) \cdot \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_1 \cdot t\_1}}\\ \end{array} \end{array} \]

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 (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_1 (/ 1.0 (* h l))))
   (if (<= t_0 -1e-90)
     (*
      (-
       1.0
       (* h (/ (* (* M D) (* D (* M 0.5))) (* (* d 2.0) (* l (* d 2.0))))))
      (sqrt (/ (/ d h) (/ l d))))
     (if (<= t_0 INFINITY)
       (/ (* d (sqrt (/ h l))) h)
       (* d (sqrt (sqrt (* t_1 t_1))))))))

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= -1e-90) {
		tmp = (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0)))))) * sqrt(((d / h) / (l / d)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	}
	return tmp;
}

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= -1e-90) {
		tmp = (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0)))))) * Math.sqrt(((d / h) / (l / d)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d * Math.sqrt(Math.sqrt((t_1 * t_1)));
	}
	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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	t_1 = 1.0 / (h * l)
	tmp = 0
	if t_0 <= -1e-90:
		tmp = (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0)))))) * math.sqrt(((d / h) / (l / d)))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = d * math.sqrt(math.sqrt((t_1 * t_1)))
	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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_1 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (t_0 <= -1e-90)
		tmp = Float64(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D) * Float64(D * Float64(M * 0.5))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0)))))) * sqrt(Float64(Float64(d / h) / Float64(l / d))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d * sqrt(sqrt(Float64(t_1 * t_1))));
	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) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	t_1 = 1.0 / (h * l);
	tmp = 0.0;
	if (t_0 <= -1e-90)
		tmp = (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0)))))) * sqrt(((d / h) / (l / d)));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-90], N[(N[(1.0 - N[(h * N[(N[(N[(M * D), $MachinePrecision] * N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $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)))) < -9.99999999999999995e-91

   1. Initial program 85.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 89.2

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

   6. Applied rewrites 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-/r/ N/A

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

      12. remove-double-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. times-frac N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 65.6%

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

   if -9.99999999999999995e-91 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 28.2%

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

   9. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 79.7

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

   11. Applied rewrites 79.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 13.8

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

   8. Applied rewrites 13.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 13.8

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right) \cdot \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_1 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 1000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_1 \cdot t\_1}}\\ \end{array} \end{array} \]

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 (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_1 (/ 1.0 (* h l))))
   (if (<= t_0 1000000.0)
     (*
      (sqrt (/ d l))
      (*
       (- 1.0 (/ (* (* D 0.125) (* D (* h (* M M)))) (* d (* d l))))
       (sqrt (/ d h))))
     (if (<= t_0 INFINITY)
       (/ (* d (sqrt (/ h l))) h)
       (* d (sqrt (sqrt (* t_1 t_1))))))))

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= 1000000.0) {
		tmp = sqrt((d / l)) * ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * sqrt((d / h)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	}
	return tmp;
}

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= 1000000.0) {
		tmp = Math.sqrt((d / l)) * ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * Math.sqrt((d / h)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d * Math.sqrt(Math.sqrt((t_1 * t_1)));
	}
	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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	t_1 = 1.0 / (h * l)
	tmp = 0
	if t_0 <= 1000000.0:
		tmp = math.sqrt((d / l)) * ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * math.sqrt((d / h)))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = d * math.sqrt(math.sqrt((t_1 * t_1)))
	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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_1 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (t_0 <= 1000000.0)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 - Float64(Float64(Float64(D * 0.125) * Float64(D * Float64(h * Float64(M * M)))) / Float64(d * Float64(d * l)))) * sqrt(Float64(d / h))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d * sqrt(sqrt(Float64(t_1 * t_1))));
	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) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	t_1 = 1.0 / (h * l);
	tmp = 0.0;
	if (t_0 <= 1000000.0)
		tmp = sqrt((d / l)) * ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * sqrt((d / h)));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1000000.0], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(D * 0.125), $MachinePrecision] * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $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)))) < 1e6

   1. Initial program 85.6%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 39.5

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

   7. Applied rewrites 39.5%

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

   9. Applied rewrites 62.4%

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

   if 1e6 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 81.2%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 28.6%

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

   9. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 86.9

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

   11. Applied rewrites 86.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 13.8

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

   8. Applied rewrites 13.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 13.8

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 1000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_1 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_1 \cdot t\_1}}\\ \end{array} \end{array} \]

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 (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_1 (/ 1.0 (* h l))))
   (if (<= t_0 -1e-90)
     (*
      (-
       1.0
       (* h (/ (* (* M D) (* (* M D) 0.5)) (* (* d 2.0) (* l (* d 2.0))))))
      (sqrt (/ (* d d) (* h l))))
     (if (<= t_0 INFINITY)
       (/ (* d (sqrt (/ h l))) h)
       (* d (sqrt (sqrt (* t_1 t_1))))))))

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= -1e-90) {
		tmp = (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))) * sqrt(((d * d) / (h * l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	}
	return tmp;
}

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= -1e-90) {
		tmp = (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))) * Math.sqrt(((d * d) / (h * l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d * Math.sqrt(Math.sqrt((t_1 * t_1)));
	}
	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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	t_1 = 1.0 / (h * l)
	tmp = 0
	if t_0 <= -1e-90:
		tmp = (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))) * math.sqrt(((d * d) / (h * l)))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = d * math.sqrt(math.sqrt((t_1 * t_1)))
	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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_1 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (t_0 <= -1e-90)
		tmp = Float64(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * 0.5)) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0)))))) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d * sqrt(sqrt(Float64(t_1 * t_1))));
	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) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	t_1 = 1.0 / (h * l);
	tmp = 0.0;
	if (t_0 <= -1e-90)
		tmp = (1.0 - (h * (((M * D) * ((M * D) * 0.5)) / ((d * 2.0) * (l * (d * 2.0)))))) * sqrt(((d * d) / (h * l)));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-90], N[(N[(1.0 - N[(h * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $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)))) < -9.99999999999999995e-91

   1. Initial program 85.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied rewrites 88.9%

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

   6. Applied rewrites 52.4%

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

   if -9.99999999999999995e-91 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 28.2%

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

   9. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 79.7

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

   11. Applied rewrites 79.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 13.8

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

   8. Applied rewrites 13.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 13.8

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_1 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_1 \cdot t\_1}}\\ \end{array} \end{array} \]

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 (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_1 (/ 1.0 (* h l))))
   (if (<= t_0 -1e-90)
     (*
      (- 1.0 (/ (* (* D 0.125) (* D (* h (* M M)))) (* d (* d l))))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 INFINITY)
       (/ (* d (sqrt (/ h l))) h)
       (* d (sqrt (sqrt (* t_1 t_1))))))))

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= -1e-90) {
		tmp = (1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	}
	return tmp;
}

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= -1e-90) {
		tmp = (1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * Math.sqrt(((d / l) * (d / h)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d * Math.sqrt(Math.sqrt((t_1 * t_1)));
	}
	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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	t_1 = 1.0 / (h * l)
	tmp = 0
	if t_0 <= -1e-90:
		tmp = (1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * math.sqrt(((d / l) * (d / h)))
	elif t_0 <= math.inf:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = d * math.sqrt(math.sqrt((t_1 * t_1)))
	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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_1 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (t_0 <= -1e-90)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(D * 0.125) * Float64(D * Float64(h * Float64(M * M)))) / Float64(d * Float64(d * l)))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d * sqrt(sqrt(Float64(t_1 * t_1))));
	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) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	t_1 = 1.0 / (h * l);
	tmp = 0.0;
	if (t_0 <= -1e-90)
		tmp = (1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * sqrt(((d / l) * (d / h)));
	elseif (t_0 <= Inf)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-90], N[(N[(1.0 - N[(N[(N[(D * 0.125), $MachinePrecision] * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $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)))) < -9.99999999999999995e-91

   1. Initial program 85.8%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 37.7

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

   7. Applied rewrites 37.7%

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

   9. Applied rewrites 55.5%

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

   if -9.99999999999999995e-91 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 28.2%

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

   9. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 79.7

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

   11. Applied rewrites 79.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 13.8

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

   8. Applied rewrites 13.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 13.8

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_1 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_1 \cdot t\_1}}\\ \end{array} \end{array} \]

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 (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_1 (/ 1.0 (* h l))))
   (if (<= t_0 -1e-90)
     (*
      (fma (- D) (/ (* (* h 0.125) (* D (* M M))) (* d (* d l))) 1.0)
      (sqrt (/ (* d d) (* h l))))
     (if (<= t_0 INFINITY)
       (/ (* d (sqrt (/ h l))) h)
       (* d (sqrt (sqrt (* t_1 t_1))))))))

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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_1 = 1.0 / (h * l);
	double tmp;
	if (t_0 <= -1e-90) {
		tmp = fma(-D, (((h * 0.125) * (D * (M * M))) / (d * (d * l))), 1.0) * sqrt(((d * d) / (h * l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = d * sqrt(sqrt((t_1 * t_1)));
	}
	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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_1 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (t_0 <= -1e-90)
		tmp = Float64(fma(Float64(-D), Float64(Float64(Float64(h * 0.125) * Float64(D * Float64(M * M))) / Float64(d * Float64(d * l))), 1.0) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d * sqrt(sqrt(Float64(t_1 * t_1))));
	end
	return 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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-90], N[(N[((-D) * N[(N[(N[(h * 0.125), $MachinePrecision] * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $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)))) < -9.99999999999999995e-91

   1. Initial program 85.8%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 46.4%

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

   if -9.99999999999999995e-91 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 28.2%

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

   9. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 79.7

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

   11. Applied rewrites 79.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 13.8

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

   8. Applied rewrites 13.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 13.8

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 84.4%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   6. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/l* N/A

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

      14. associate-*r* N/A

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

   8. Applied rewrites 41.8%

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

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

   1. Initial program 84.2%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 25.3%

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

   9. Taylor expanded in d around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 71.6

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

   11. Applied rewrites 71.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 13.8

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

   8. Applied rewrites 13.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. lift-/.f64 13.8

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      3. rem-square-sqrt N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      4. sqrt-unprod N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]

      6. lower-*.f64 20.0

         \[\leadsto d \cdot \sqrt{\sqrt{\color{blue}{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]

   10. Applied rewrites 20.0%

       \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -\infty:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_2 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-140}:\\ \;\;\;\;\frac{t\_0}{-h}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_2 \cdot t\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ h l))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_2 (/ 1.0 (* h l))))
   (if (<= t_1 -4e-140)
     (/ t_0 (- h))
     (if (<= t_1 INFINITY) (/ t_0 h) (* d (sqrt (sqrt (* t_2 t_2))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_2 = 1.0 / (h * l);
	double tmp;
	if (t_1 <= -4e-140) {
		tmp = t_0 / -h;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0 / h;
	} else {
		tmp = d * sqrt(sqrt((t_2 * t_2)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_2 = 1.0 / (h * l);
	double tmp;
	if (t_1 <= -4e-140) {
		tmp = t_0 / -h;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / h;
	} else {
		tmp = d * Math.sqrt(Math.sqrt((t_2 * t_2)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	t_2 = 1.0 / (h * l)
	tmp = 0
	if t_1 <= -4e-140:
		tmp = t_0 / -h
	elif t_1 <= math.inf:
		tmp = t_0 / h
	else:
		tmp = d * math.sqrt(math.sqrt((t_2 * t_2)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * sqrt(Float64(h / l)))
	t_1 = 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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_2 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (t_1 <= -4e-140)
		tmp = Float64(t_0 / Float64(-h));
	elseif (t_1 <= Inf)
		tmp = Float64(t_0 / h);
	else
		tmp = Float64(d * sqrt(sqrt(Float64(t_2 * t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	t_2 = 1.0 / (h * l);
	tmp = 0.0;
	if (t_1 <= -4e-140)
		tmp = t_0 / -h;
	elseif (t_1 <= Inf)
		tmp = t_0 / h;
	else
		tmp = d * sqrt(sqrt((t_2 * t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-140], N[(t$95$0 / (-h)), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$0 / h), $MachinePrecision], N[(d * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $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)))) < -3.9999999999999999e-140

   1. Initial program 85.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 60.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 60.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Applied rewrites 27.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(-0.125 \cdot \frac{M \cdot M}{d}\right)\right)\right)}{h}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \frac{\color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}}{h} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}{h} \]

       2. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}}{h} \]

       3. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)}{h} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right)}{h} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}}{h} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}}} \cdot \left(-1 \cdot d\right)}{h} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{h}{\ell}}} \cdot \left(-1 \cdot d\right)}{h} \]

       8. mul-1-neg N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}{h} \]

       9. lower-neg.f64 25.1

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(-d\right)}}{h} \]

   11. Applied rewrites 25.1%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}}{h} \]

   if -3.9999999999999999e-140 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 83.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 63.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 63.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Applied rewrites 28.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(-0.125 \cdot \frac{M \cdot M}{d}\right)\right)\right)}{h}} \]

   9. Taylor expanded in d around inf

      \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \color{blue}{\sqrt{\frac{h}{\ell}}}}{h} \]

       3. lower-/.f64 80.4

          \[\leadsto \frac{d \cdot \sqrt{\color{blue}{\frac{h}{\ell}}}}{h} \]

   11. Applied rewrites 80.4%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 12.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 12.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 13.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 13.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. lift-/.f64 13.8

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      3. rem-square-sqrt N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      4. sqrt-unprod N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]

      6. lower-*.f64 20.0

         \[\leadsto d \cdot \sqrt{\sqrt{\color{blue}{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]

   10. Applied rewrites 20.0%

       \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -4 \cdot 10^{-140}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{-h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \sqrt{\frac{h}{\ell}}\\ t_1 := \frac{t\_0}{-h}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ h l))))
        (t_1 (/ t_0 (- h)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_2 -4e-140) t_1 (if (<= t_2 INFINITY) (/ t_0 h) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * sqrt((h / l));
	double t_1 = t_0 / -h;
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_2 <= -4e-140) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 / h;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.sqrt((h / l));
	double t_1 = t_0 / -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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_2 <= -4e-140) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / h;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.sqrt((h / l))
	t_1 = t_0 / -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(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_2 <= -4e-140:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = t_0 / h
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * sqrt(Float64(h / l)))
	t_1 = Float64(t_0 / Float64(-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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_2 <= -4e-140)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 / h);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * sqrt((h / l));
	t_1 = t_0 / -h;
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_2 <= -4e-140)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = t_0 / h;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / (-h)), $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[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-140], t$95$1, If[LessEqual[t$95$2, Infinity], N[(t$95$0 / h), $MachinePrecision], t$95$1]]]]]

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)))) < -3.9999999999999999e-140 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 44.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 44.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Applied rewrites 19.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(-0.125 \cdot \frac{M \cdot M}{d}\right)\right)\right)}{h}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \frac{\color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}}{h} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}{h} \]

       2. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}}{h} \]

       3. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)}{h} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right)}{h} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}}{h} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}}} \cdot \left(-1 \cdot d\right)}{h} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{h}{\ell}}} \cdot \left(-1 \cdot d\right)}{h} \]

       8. mul-1-neg N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}{h} \]

       9. lower-neg.f64 22.7

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(-d\right)}}{h} \]

   11. Applied rewrites 22.7%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}}{h} \]

   if -3.9999999999999999e-140 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 83.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 63.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 63.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Applied rewrites 28.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(-0.125 \cdot \frac{M \cdot M}{d}\right)\right)\right)}{h}} \]

   9. Taylor expanded in d around inf

      \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \color{blue}{\sqrt{\frac{h}{\ell}}}}{h} \]

       3. lower-/.f64 80.4

          \[\leadsto \frac{d \cdot \sqrt{\color{blue}{\frac{h}{\ell}}}}{h} \]

   11. Applied rewrites 80.4%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -4 \cdot 10^{-140}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{-h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{-h}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}\\ t_1 := \frac{\frac{M \cdot D}{d \cdot 2}}{\ell}\\ t_2 := 1 + t\_1 \cdot \frac{t\_0}{\frac{-1}{h}}\\ \mathbf{if}\;h \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;t\_2 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t\_1 \cdot \frac{t\_0}{\frac{1}{h}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* (* M D) 0.5) (* d 2.0)))
        (t_1 (/ (/ (* M D) (* d 2.0)) l))
        (t_2 (+ 1.0 (* t_1 (/ t_0 (/ -1.0 h))))))
   (if (<= h -3.2e+170)
     (* t_2 (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))))
     (if (<= h -5e-310)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ (* t_1 (/ t_0 (/ 1.0 h))) -1.0))
       (* t_2 (* (* (sqrt (/ 1.0 h)) (sqrt d)) (/ 1.0 (sqrt (/ l d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) * 0.5) / (d * 2.0);
	double t_1 = ((M * D) / (d * 2.0)) / l;
	double t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)));
	double tmp;
	if (h <= -3.2e+170) {
		tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0);
	} else {
		tmp = t_2 * ((sqrt((1.0 / h)) * sqrt(d)) * (1.0 / sqrt((l / d))));
	}
	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 * d_1) * 0.5d0) / (d * 2.0d0)
    t_1 = ((m * d_1) / (d * 2.0d0)) / l
    t_2 = 1.0d0 + (t_1 * (t_0 / ((-1.0d0) / h)))
    if (h <= (-3.2d+170)) then
        tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)))
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((t_1 * (t_0 / (1.0d0 / h))) + (-1.0d0))
    else
        tmp = t_2 * ((sqrt((1.0d0 / h)) * sqrt(d)) * (1.0d0 / sqrt((l / d))))
    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) * 0.5) / (d * 2.0);
	double t_1 = ((M * D) / (d * 2.0)) / l;
	double t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)));
	double tmp;
	if (h <= -3.2e+170) {
		tmp = t_2 * ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l)));
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0);
	} else {
		tmp = t_2 * ((Math.sqrt((1.0 / h)) * Math.sqrt(d)) * (1.0 / Math.sqrt((l / d))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = ((M * D) * 0.5) / (d * 2.0)
	t_1 = ((M * D) / (d * 2.0)) / l
	t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)))
	tmp = 0
	if h <= -3.2e+170:
		tmp = t_2 * ((math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l)))
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0)
	else:
		tmp = t_2 * ((math.sqrt((1.0 / h)) * math.sqrt(d)) * (1.0 / math.sqrt((l / d))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(M * D) * 0.5) / Float64(d * 2.0))
	t_1 = Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / l)
	t_2 = Float64(1.0 + Float64(t_1 * Float64(t_0 / Float64(-1.0 / h))))
	tmp = 0.0
	if (h <= -3.2e+170)
		tmp = Float64(t_2 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(t_1 * Float64(t_0 / Float64(1.0 / h))) + -1.0));
	else
		tmp = Float64(t_2 * Float64(Float64(sqrt(Float64(1.0 / h)) * sqrt(d)) * Float64(1.0 / sqrt(Float64(l / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) * 0.5) / (d * 2.0);
	t_1 = ((M * D) / (d * 2.0)) / l;
	t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)));
	tmp = 0.0;
	if (h <= -3.2e+170)
		tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0);
	else
		tmp = t_2 * ((sqrt((1.0 / h)) * sqrt(d)) * (1.0 / sqrt((l / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(t$95$1 * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.2e+170], N[(t$95$2 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(t$95$0 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.19999999999999979e170

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. lower-neg.f64 90.0

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 90.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lower-sqrt.f64 90.0

         \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 90.0%

      \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if -3.19999999999999979e170 < h < -4.999999999999985e-310

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 74.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 74.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 74.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 0.0

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       9. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       10. lower-neg.f64 85.9

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 85.9%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if -4.999999999999985e-310 < h

   1. Initial program 72.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 74.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 76.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 76.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 84.9

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 84.9%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{1}{h}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}\\ t_1 := \frac{\frac{M \cdot D}{d \cdot 2}}{\ell}\\ t_2 := 1 + t\_1 \cdot \frac{t\_0}{\frac{-1}{h}}\\ \mathbf{if}\;h \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;t\_2 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t\_1 \cdot \frac{t\_0}{\frac{1}{h}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* (* M D) 0.5) (* d 2.0)))
        (t_1 (/ (/ (* M D) (* d 2.0)) l))
        (t_2 (+ 1.0 (* t_1 (/ t_0 (/ -1.0 h))))))
   (if (<= h -3.2e+170)
     (* t_2 (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))))
     (if (<= h -5e-310)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ (* t_1 (/ t_0 (/ 1.0 h))) -1.0))
       (* t_2 (* (/ 1.0 (sqrt (/ l d))) (/ (sqrt d) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) * 0.5) / (d * 2.0);
	double t_1 = ((M * D) / (d * 2.0)) / l;
	double t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)));
	double tmp;
	if (h <= -3.2e+170) {
		tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0);
	} else {
		tmp = t_2 * ((1.0 / sqrt((l / 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 * d_1) * 0.5d0) / (d * 2.0d0)
    t_1 = ((m * d_1) / (d * 2.0d0)) / l
    t_2 = 1.0d0 + (t_1 * (t_0 / ((-1.0d0) / h)))
    if (h <= (-3.2d+170)) then
        tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)))
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((t_1 * (t_0 / (1.0d0 / h))) + (-1.0d0))
    else
        tmp = t_2 * ((1.0d0 / sqrt((l / 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 * D) * 0.5) / (d * 2.0);
	double t_1 = ((M * D) / (d * 2.0)) / l;
	double t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)));
	double tmp;
	if (h <= -3.2e+170) {
		tmp = t_2 * ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l)));
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0);
	} else {
		tmp = t_2 * ((1.0 / Math.sqrt((l / d))) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = ((M * D) * 0.5) / (d * 2.0)
	t_1 = ((M * D) / (d * 2.0)) / l
	t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)))
	tmp = 0
	if h <= -3.2e+170:
		tmp = t_2 * ((math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l)))
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0)
	else:
		tmp = t_2 * ((1.0 / math.sqrt((l / d))) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(M * D) * 0.5) / Float64(d * 2.0))
	t_1 = Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / l)
	t_2 = Float64(1.0 + Float64(t_1 * Float64(t_0 / Float64(-1.0 / h))))
	tmp = 0.0
	if (h <= -3.2e+170)
		tmp = Float64(t_2 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(t_1 * Float64(t_0 / Float64(1.0 / h))) + -1.0));
	else
		tmp = Float64(t_2 * Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) * 0.5) / (d * 2.0);
	t_1 = ((M * D) / (d * 2.0)) / l;
	t_2 = 1.0 + (t_1 * (t_0 / (-1.0 / h)));
	tmp = 0.0;
	if (h <= -3.2e+170)
		tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_1 * (t_0 / (1.0 / h))) + -1.0);
	else
		tmp = t_2 * ((1.0 / sqrt((l / d))) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(t$95$1 * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.2e+170], N[(t$95$2 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(t$95$0 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.19999999999999979e170

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. lower-neg.f64 90.0

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 90.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lower-sqrt.f64 90.0

         \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 90.0%

      \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if -3.19999999999999979e170 < h < -4.999999999999985e-310

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 74.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 74.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 74.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 0.0

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       9. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       10. lower-neg.f64 85.9

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 85.9%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if -4.999999999999985e-310 < h

   1. Initial program 72.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 74.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 76.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 76.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \left(\frac{{d}^{\left(\frac{1}{2}\right)}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{{d}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\frac{{d}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{{d}^{\left(\frac{1}{2}\right)}}{{h}^{\left(\frac{1}{2}\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{{h}^{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\frac{1}{2}}}}{{h}^{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{{h}^{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{{h}^{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\frac{\sqrt{d}}{{h}^{\color{blue}{\frac{1}{2}}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 84.9

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 84.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{1}{h}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}\\ t_1 := d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := \frac{\frac{M \cdot D}{d \cdot 2}}{\ell}\\ t_3 := 1 + t\_2 \cdot \frac{t\_0}{\frac{-1}{h}}\\ \mathbf{if}\;h \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;t\_3 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \frac{t\_0}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;h \leq 5 \cdot 10^{+106}:\\ \;\;\;\;t\_3 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - t\_2 \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot 0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* (* M D) 0.5) (* d 2.0)))
        (t_1 (* d (sqrt (/ 1.0 (* h l)))))
        (t_2 (/ (/ (* M D) (* d 2.0)) l))
        (t_3 (+ 1.0 (* t_2 (/ t_0 (/ -1.0 h))))))
   (if (<= h -3.2e+170)
     (* t_3 (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))))
     (if (<= h -5e-310)
       (* t_1 (+ (* t_2 (/ t_0 (/ 1.0 h))) -1.0))
       (if (<= h 5e+106)
         (* t_3 t_1)
         (*
          (* (* (sqrt (/ 1.0 h)) (sqrt d)) (/ 1.0 (sqrt (/ l d))))
          (- 1.0 (* t_2 (* D (* (/ (* h M) d) 0.25))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) * 0.5) / (d * 2.0);
	double t_1 = d * sqrt((1.0 / (h * l)));
	double t_2 = ((M * D) / (d * 2.0)) / l;
	double t_3 = 1.0 + (t_2 * (t_0 / (-1.0 / h)));
	double tmp;
	if (h <= -3.2e+170) {
		tmp = t_3 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	} else if (h <= -5e-310) {
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0);
	} else if (h <= 5e+106) {
		tmp = t_3 * t_1;
	} else {
		tmp = ((sqrt((1.0 / h)) * sqrt(d)) * (1.0 / sqrt((l / d)))) * (1.0 - (t_2 * (D * (((h * M) / d) * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ((m * d_1) * 0.5d0) / (d * 2.0d0)
    t_1 = d * sqrt((1.0d0 / (h * l)))
    t_2 = ((m * d_1) / (d * 2.0d0)) / l
    t_3 = 1.0d0 + (t_2 * (t_0 / ((-1.0d0) / h)))
    if (h <= (-3.2d+170)) then
        tmp = t_3 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)))
    else if (h <= (-5d-310)) then
        tmp = t_1 * ((t_2 * (t_0 / (1.0d0 / h))) + (-1.0d0))
    else if (h <= 5d+106) then
        tmp = t_3 * t_1
    else
        tmp = ((sqrt((1.0d0 / h)) * sqrt(d)) * (1.0d0 / sqrt((l / d)))) * (1.0d0 - (t_2 * (d_1 * (((h * m) / d) * 0.25d0))))
    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) * 0.5) / (d * 2.0);
	double t_1 = d * Math.sqrt((1.0 / (h * l)));
	double t_2 = ((M * D) / (d * 2.0)) / l;
	double t_3 = 1.0 + (t_2 * (t_0 / (-1.0 / h)));
	double tmp;
	if (h <= -3.2e+170) {
		tmp = t_3 * ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l)));
	} else if (h <= -5e-310) {
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0);
	} else if (h <= 5e+106) {
		tmp = t_3 * t_1;
	} else {
		tmp = ((Math.sqrt((1.0 / h)) * Math.sqrt(d)) * (1.0 / Math.sqrt((l / d)))) * (1.0 - (t_2 * (D * (((h * M) / d) * 0.25))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = ((M * D) * 0.5) / (d * 2.0)
	t_1 = d * math.sqrt((1.0 / (h * l)))
	t_2 = ((M * D) / (d * 2.0)) / l
	t_3 = 1.0 + (t_2 * (t_0 / (-1.0 / h)))
	tmp = 0
	if h <= -3.2e+170:
		tmp = t_3 * ((math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l)))
	elif h <= -5e-310:
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0)
	elif h <= 5e+106:
		tmp = t_3 * t_1
	else:
		tmp = ((math.sqrt((1.0 / h)) * math.sqrt(d)) * (1.0 / math.sqrt((l / d)))) * (1.0 - (t_2 * (D * (((h * M) / d) * 0.25))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(M * D) * 0.5) / Float64(d * 2.0))
	t_1 = Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
	t_2 = Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / l)
	t_3 = Float64(1.0 + Float64(t_2 * Float64(t_0 / Float64(-1.0 / h))))
	tmp = 0.0
	if (h <= -3.2e+170)
		tmp = Float64(t_3 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (h <= -5e-310)
		tmp = Float64(t_1 * Float64(Float64(t_2 * Float64(t_0 / Float64(1.0 / h))) + -1.0));
	elseif (h <= 5e+106)
		tmp = Float64(t_3 * t_1);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / h)) * sqrt(d)) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(t_2 * Float64(D * Float64(Float64(Float64(h * M) / d) * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) * 0.5) / (d * 2.0);
	t_1 = d * sqrt((1.0 / (h * l)));
	t_2 = ((M * D) / (d * 2.0)) / l;
	t_3 = 1.0 + (t_2 * (t_0 / (-1.0 / h)));
	tmp = 0.0;
	if (h <= -3.2e+170)
		tmp = t_3 * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	elseif (h <= -5e-310)
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0);
	elseif (h <= 5e+106)
		tmp = t_3 * t_1;
	else
		tmp = ((sqrt((1.0 / h)) * sqrt(d)) * (1.0 / sqrt((l / d)))) * (1.0 - (t_2 * (D * (((h * M) / d) * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(t$95$2 * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.2e+170], N[(t$95$3 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(t$95$1 * N[(N[(t$95$2 * N[(t$95$0 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5e+106], N[(t$95$3 * t$95$1), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -3.19999999999999979e170

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. lower-neg.f64 90.0

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 90.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lower-sqrt.f64 90.0

         \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 90.0%

      \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if -3.19999999999999979e170 < h < -4.999999999999985e-310

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 74.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 74.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 74.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 0.0

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       9. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       10. lower-neg.f64 85.9

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 85.9%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if -4.999999999999985e-310 < h < 4.9999999999999998e106

   1. Initial program 75.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 78.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 80.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 80.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 87.2

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 87.2%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. lower-*.f64 84.8

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 84.8%

       \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if 4.9999999999999998e106 < h

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 66.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 68.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 68.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 80.5

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 80.5%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in M around 0

      \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{D \cdot \left(M \cdot h\right)}{d} \cdot \frac{1}{4}\right)}\right) \]

       2. associate-/l* N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\left(D \cdot \frac{M \cdot h}{d}\right)} \cdot \frac{1}{4}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(D \cdot \left(\frac{M \cdot h}{d} \cdot \frac{1}{4}\right)\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(D \cdot \left(\frac{M \cdot h}{d} \cdot \frac{1}{4}\right)\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(D \cdot \color{blue}{\left(\frac{M \cdot h}{d} \cdot \frac{1}{4}\right)}\right)\right) \]

       6. lower-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(D \cdot \left(\color{blue}{\frac{M \cdot h}{d}} \cdot \frac{1}{4}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(D \cdot \left(\frac{\color{blue}{h \cdot M}}{d} \cdot \frac{1}{4}\right)\right)\right) \]

       8. lower-*.f64 71.0

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(D \cdot \left(\frac{\color{blue}{h \cdot M}}{d} \cdot 0.25\right)\right)\right) \]

   11. Applied rewrites 71.0%

       \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(D \cdot \left(\frac{h \cdot M}{d} \cdot 0.25\right)\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;h \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{M \cdot D}{d \cdot 2}}{\ell}\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{d \cdot \frac{d}{-h}}\right) \cdot \sqrt{\frac{-1}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - t\_0 \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + t\_0 \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (/ (* M D) (* d 2.0)) l)))
   (if (<= l -2.2e+228)
     (*
      (*
       (- 1.0 (/ (* (* D 0.125) (* D (* h (* M M)))) (* d (* d l))))
       (sqrt (* d (/ d (- h)))))
      (sqrt (/ -1.0 l)))
     (if (<= l -5e-311)
       (*
        (* (/ (sqrt (- d)) (sqrt (- h))) (/ 1.0 (sqrt (/ l d))))
        (- 1.0 (* t_0 (/ h (* (/ d (* M D)) 4.0)))))
       (if (<= l 2.2e+94)
         (*
          (+ 1.0 (* t_0 (/ (/ (* (* M D) 0.5) (* d 2.0)) (/ -1.0 h))))
          (* d (sqrt (/ 1.0 (* h l)))))
         (/ (/ d (sqrt h)) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) / (d * 2.0)) / l;
	double tmp;
	if (l <= -2.2e+228) {
		tmp = ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * sqrt((d * (d / -h)))) * sqrt((-1.0 / l));
	} else if (l <= -5e-311) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (t_0 * (h / ((d / (M * D)) * 4.0))));
	} else if (l <= 2.2e+94) {
		tmp = (1.0 + (t_0 * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m * d_1) / (d * 2.0d0)) / l
    if (l <= (-2.2d+228)) then
        tmp = ((1.0d0 - (((d_1 * 0.125d0) * (d_1 * (h * (m * m)))) / (d * (d * l)))) * sqrt((d * (d / -h)))) * sqrt(((-1.0d0) / l))
    else if (l <= (-5d-311)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 / sqrt((l / d)))) * (1.0d0 - (t_0 * (h / ((d / (m * d_1)) * 4.0d0))))
    else if (l <= 2.2d+94) then
        tmp = (1.0d0 + (t_0 * ((((m * d_1) * 0.5d0) / (d * 2.0d0)) / ((-1.0d0) / h)))) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) / (d * 2.0)) / l;
	double tmp;
	if (l <= -2.2e+228) {
		tmp = ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * Math.sqrt((d * (d / -h)))) * Math.sqrt((-1.0 / l));
	} else if (l <= -5e-311) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 / Math.sqrt((l / d)))) * (1.0 - (t_0 * (h / ((d / (M * D)) * 4.0))));
	} else if (l <= 2.2e+94) {
		tmp = (1.0 + (t_0 * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = ((M * D) / (d * 2.0)) / l
	tmp = 0
	if l <= -2.2e+228:
		tmp = ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * math.sqrt((d * (d / -h)))) * math.sqrt((-1.0 / l))
	elif l <= -5e-311:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 / math.sqrt((l / d)))) * (1.0 - (t_0 * (h / ((d / (M * D)) * 4.0))))
	elif l <= 2.2e+94:
		tmp = (1.0 + (t_0 * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / l)
	tmp = 0.0
	if (l <= -2.2e+228)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(D * 0.125) * Float64(D * Float64(h * Float64(M * M)))) / Float64(d * Float64(d * l)))) * sqrt(Float64(d * Float64(d / Float64(-h))))) * sqrt(Float64(-1.0 / l)));
	elseif (l <= -5e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(t_0 * Float64(h / Float64(Float64(d / Float64(M * D)) * 4.0)))));
	elseif (l <= 2.2e+94)
		tmp = Float64(Float64(1.0 + Float64(t_0 * Float64(Float64(Float64(Float64(M * D) * 0.5) / Float64(d * 2.0)) / Float64(-1.0 / h)))) * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) / (d * 2.0)) / l;
	tmp = 0.0;
	if (l <= -2.2e+228)
		tmp = ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / (d * (d * l)))) * sqrt((d * (d / -h)))) * sqrt((-1.0 / l));
	elseif (l <= -5e-311)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (t_0 * (h / ((d / (M * D)) * 4.0))));
	elseif (l <= 2.2e+94)
		tmp = (1.0 + (t_0 * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[l, -2.2e+228], N[(N[(N[(1.0 - N[(N[(N[(D * 0.125), $MachinePrecision] * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d * N[(d / (-h)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(h / N[(N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+94], N[(N[(1.0 + N[(t$95$0 * N[(N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.2e228

   1. Initial program 44.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 38.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 38.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-neg.f64 68.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-\ell}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied rewrites 68.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   9. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \left(-d\right)}\right) \cdot \sqrt{\frac{-1}{\ell}}} \]

   if -2.2e228 < l < -5.00000000000023e-311

   1. Initial program 70.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 76.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 76.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 76.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      5. associate-/r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{1} \cdot h\right)}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\frac{1}{\frac{1}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      8. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\color{blue}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}} \cdot h\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{1 \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{1}} \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      11. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{\frac{1}{h}}}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      12. remove-double-div N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{h}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{\color{blue}{d \cdot 2}}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      18. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}}\right) \]

      19. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D}} \cdot \frac{2}{\frac{1}{2}}}\right) \]

   8. Applied rewrites 76.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d}{M \cdot D} \cdot 4}}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

      11. lower-neg.f64 87.1

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

   10. Applied rewrites 87.1%

       \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right) \]

   if -5.00000000000023e-311 < l < 2.20000000000000012e94

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 79.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 82.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 82.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 91.5

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 91.5%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. lower-*.f64 87.9

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 87.9%

       \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if 2.20000000000000012e94 < l

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 47.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 47.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 49.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 49.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-sqrt.f64 49.8

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 49.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       2. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{h}}}}{\sqrt{\ell}} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

       6. lower-sqrt.f64 77.3

          \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   12. Applied rewrites 77.3%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{d \cdot \frac{d}{-h}}\right) \cdot \sqrt{\frac{-1}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d}{M \cdot D} \cdot 4}\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}\\ t_1 := d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := \frac{\frac{M \cdot D}{d \cdot 2}}{\ell}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+143}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \frac{t\_0}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + t\_2 \cdot \frac{t\_0}{\frac{-1}{h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* (* M D) 0.5) (* d 2.0)))
        (t_1 (* d (sqrt (/ 1.0 (* h l)))))
        (t_2 (/ (/ (* M D) (* d 2.0)) l)))
   (if (<= l -3.6e+143)
     (*
      (* (sqrt (/ d h)) (* (sqrt (- d)) (sqrt (/ -1.0 l))))
      (- 1.0 (/ (* D (* D (* 0.125 (* h (* M M))))) (* d (* d l)))))
     (if (<= l -5e-311)
       (* t_1 (+ (* t_2 (/ t_0 (/ 1.0 h))) -1.0))
       (if (<= l 2.2e+94)
         (* (+ 1.0 (* t_2 (/ t_0 (/ -1.0 h)))) t_1)
         (/ (/ d (sqrt h)) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) * 0.5) / (d * 2.0);
	double t_1 = d * sqrt((1.0 / (h * l)));
	double t_2 = ((M * D) / (d * 2.0)) / l;
	double tmp;
	if (l <= -3.6e+143) {
		tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	} else if (l <= -5e-311) {
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0);
	} else if (l <= 2.2e+94) {
		tmp = (1.0 + (t_2 * (t_0 / (-1.0 / h)))) * t_1;
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((m * d_1) * 0.5d0) / (d * 2.0d0)
    t_1 = d * sqrt((1.0d0 / (h * l)))
    t_2 = ((m * d_1) / (d * 2.0d0)) / l
    if (l <= (-3.6d+143)) then
        tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt(((-1.0d0) / l)))) * (1.0d0 - ((d_1 * (d_1 * (0.125d0 * (h * (m * m))))) / (d * (d * l))))
    else if (l <= (-5d-311)) then
        tmp = t_1 * ((t_2 * (t_0 / (1.0d0 / h))) + (-1.0d0))
    else if (l <= 2.2d+94) then
        tmp = (1.0d0 + (t_2 * (t_0 / ((-1.0d0) / h)))) * t_1
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) * 0.5) / (d * 2.0);
	double t_1 = d * Math.sqrt((1.0 / (h * l)));
	double t_2 = ((M * D) / (d * 2.0)) / l;
	double tmp;
	if (l <= -3.6e+143) {
		tmp = (Math.sqrt((d / h)) * (Math.sqrt(-d) * Math.sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	} else if (l <= -5e-311) {
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0);
	} else if (l <= 2.2e+94) {
		tmp = (1.0 + (t_2 * (t_0 / (-1.0 / h)))) * t_1;
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = ((M * D) * 0.5) / (d * 2.0)
	t_1 = d * math.sqrt((1.0 / (h * l)))
	t_2 = ((M * D) / (d * 2.0)) / l
	tmp = 0
	if l <= -3.6e+143:
		tmp = (math.sqrt((d / h)) * (math.sqrt(-d) * math.sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))))
	elif l <= -5e-311:
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0)
	elif l <= 2.2e+94:
		tmp = (1.0 + (t_2 * (t_0 / (-1.0 / h)))) * t_1
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(M * D) * 0.5) / Float64(d * 2.0))
	t_1 = Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
	t_2 = Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / l)
	tmp = 0.0
	if (l <= -3.6e+143)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * Float64(1.0 - Float64(Float64(D * Float64(D * Float64(0.125 * Float64(h * Float64(M * M))))) / Float64(d * Float64(d * l)))));
	elseif (l <= -5e-311)
		tmp = Float64(t_1 * Float64(Float64(t_2 * Float64(t_0 / Float64(1.0 / h))) + -1.0));
	elseif (l <= 2.2e+94)
		tmp = Float64(Float64(1.0 + Float64(t_2 * Float64(t_0 / Float64(-1.0 / h)))) * t_1);
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) * 0.5) / (d * 2.0);
	t_1 = d * sqrt((1.0 / (h * l)));
	t_2 = ((M * D) / (d * 2.0)) / l;
	tmp = 0.0;
	if (l <= -3.6e+143)
		tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	elseif (l <= -5e-311)
		tmp = t_1 * ((t_2 * (t_0 / (1.0 / h))) + -1.0);
	elseif (l <= 2.2e+94)
		tmp = (1.0 + (t_2 * (t_0 / (-1.0 / h)))) * t_1;
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[l, -3.6e+143], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(D * N[(D * N[(0.125 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-311], N[(t$95$1 * N[(N[(t$95$2 * N[(t$95$0 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+94], N[(N[(1.0 + N[(t$95$2 * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.5999999999999999e143

   1. Initial program 55.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 47.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 47.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-neg.f64 65.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-\ell}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied rewrites 65.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lower-sqrt.f64 65.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   9. Applied rewrites 65.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   if -3.5999999999999999e143 < l < -5.00000000000023e-311

   1. Initial program 72.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 78.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 78.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 78.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 0.0

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       9. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       10. lower-neg.f64 89.1

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 89.1%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if -5.00000000000023e-311 < l < 2.20000000000000012e94

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 79.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 82.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 82.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 91.5

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 91.5%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. lower-*.f64 87.9

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 87.9%

       \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if 2.20000000000000012e94 < l

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 47.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 47.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 49.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 49.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-sqrt.f64 49.8

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 49.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       2. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{h}}}}{\sqrt{\ell}} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

       6. lower-sqrt.f64 77.3

          \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   12. Applied rewrites 77.3%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+143}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+45}:\\ \;\;\;\;\left(t\_0 \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{t\_0 \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= l -5.1e+45)
     (*
      (* t_0 (* (sqrt (- d)) (sqrt (/ -1.0 l))))
      (- 1.0 (/ (* D (* D (* 0.125 (* h (* M M))))) (* d (* d l)))))
     (if (<= l -5e-311)
       (/
        (*
         t_0
         (-
          1.0
          (* h (/ (* (* M D) (* D (* M 0.5))) (* (* d 2.0) (* l (* d 2.0)))))))
        (sqrt (/ l d)))
       (if (<= l 2.2e+94)
         (*
          (+
           1.0
           (*
            (/ (/ (* M D) (* d 2.0)) l)
            (/ (/ (* (* M D) 0.5) (* d 2.0)) (/ -1.0 h))))
          (* d (sqrt (/ 1.0 (* h l)))))
         (/ (/ d (sqrt h)) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (l <= -5.1e+45) {
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	} else if (l <= -5e-311) {
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt((l / d));
	} else if (l <= 2.2e+94) {
		tmp = (1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / h))
    if (l <= (-5.1d+45)) then
        tmp = (t_0 * (sqrt(-d) * sqrt(((-1.0d0) / l)))) * (1.0d0 - ((d_1 * (d_1 * (0.125d0 * (h * (m * m))))) / (d * (d * l))))
    else if (l <= (-5d-311)) then
        tmp = (t_0 * (1.0d0 - (h * (((m * d_1) * (d_1 * (m * 0.5d0))) / ((d * 2.0d0) * (l * (d * 2.0d0))))))) / sqrt((l / d))
    else if (l <= 2.2d+94) then
        tmp = (1.0d0 + ((((m * d_1) / (d * 2.0d0)) / l) * ((((m * d_1) * 0.5d0) / (d * 2.0d0)) / ((-1.0d0) / h)))) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double tmp;
	if (l <= -5.1e+45) {
		tmp = (t_0 * (Math.sqrt(-d) * Math.sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	} else if (l <= -5e-311) {
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / Math.sqrt((l / d));
	} else if (l <= 2.2e+94) {
		tmp = (1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	tmp = 0
	if l <= -5.1e+45:
		tmp = (t_0 * (math.sqrt(-d) * math.sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))))
	elif l <= -5e-311:
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / math.sqrt((l / d))
	elif l <= 2.2e+94:
		tmp = (1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -5.1e+45)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * Float64(1.0 - Float64(Float64(D * Float64(D * Float64(0.125 * Float64(h * Float64(M * M))))) / Float64(d * Float64(d * l)))));
	elseif (l <= -5e-311)
		tmp = Float64(Float64(t_0 * Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D) * Float64(D * Float64(M * 0.5))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0))))))) / sqrt(Float64(l / d)));
	elseif (l <= 2.2e+94)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / l) * Float64(Float64(Float64(Float64(M * D) * 0.5) / Float64(d * 2.0)) / Float64(-1.0 / h)))) * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	tmp = 0.0;
	if (l <= -5.1e+45)
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	elseif (l <= -5e-311)
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt((l / d));
	elseif (l <= 2.2e+94)
		tmp = (1.0 + ((((M * D) / (d * 2.0)) / l) * ((((M * D) * 0.5) / (d * 2.0)) / (-1.0 / h)))) * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.1e+45], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(D * N[(D * N[(0.125 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-311], N[(N[(t$95$0 * N[(1.0 - N[(h * N[(N[(N[(M * D), $MachinePrecision] * N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+94], N[(N[(1.0 + N[(N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.0999999999999997e45

   1. Initial program 54.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 48.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 48.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-neg.f64 62.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-\ell}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied rewrites 62.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lower-sqrt.f64 62.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   9. Applied rewrites 62.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   if -5.0999999999999997e45 < l < -5.00000000000023e-311

   1. Initial program 76.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 81.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 81.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 81.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      5. associate-/r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{1} \cdot h\right)}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\frac{1}{\frac{1}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      8. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\color{blue}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}} \cdot h\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{1 \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{1}} \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      11. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{\frac{1}{h}}}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      12. remove-double-div N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{h}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{\color{blue}{d \cdot 2}}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      18. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}}\right) \]

      19. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D}} \cdot \frac{2}{\frac{1}{2}}}\right) \]

   8. Applied rewrites 81.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d}{M \cdot D} \cdot 4}}\right) \]

   10. Applied rewrites 65.8%

       \[\leadsto \color{blue}{\frac{\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}} \]

   if -5.00000000000023e-311 < l < 2.20000000000000012e94

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 79.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 82.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 82.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h} \cdot d\right)}}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\color{blue}{\frac{1}{h}} \cdot d\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{1}{h} \cdot d\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left({\left(\frac{1}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. pow1/2 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      11. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left({h}^{\left(\frac{-1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      13. sqrt-pow1 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      14. inv-pow N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{h}}} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot {d}^{\color{blue}{\frac{1}{2}}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      20. lower-sqrt.f64 91.5

          \[\leadsto \left(\left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied rewrites 91.5%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \sqrt{d}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

       4. lower-*.f64 87.9

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   11. Applied rewrites 87.9%

       \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   if 2.20000000000000012e94 < l

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 47.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 47.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 49.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 49.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-sqrt.f64 49.8

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 49.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       2. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{h}}}}{\sqrt{\ell}} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

       6. lower-sqrt.f64 77.3

          \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   12. Applied rewrites 77.3%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+45}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+45}:\\ \;\;\;\;\left(t\_0 \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{-86}:\\ \;\;\;\;\frac{t\_0 \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= l -5.1e+45)
     (*
      (* t_0 (* (sqrt (- d)) (sqrt (/ -1.0 l))))
      (- 1.0 (/ (* D (* D (* 0.125 (* h (* M M))))) (* d (* d l)))))
     (if (<= l 1.52e-86)
       (/
        (*
         t_0
         (-
          1.0
          (* h (/ (* (* M D) (* D (* M 0.5))) (* (* d 2.0) (* l (* d 2.0)))))))
        (sqrt (/ l d)))
       (* d (/ (/ 1.0 (sqrt l)) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (l <= -5.1e+45) {
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	} else if (l <= 1.52e-86) {
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt((l / d));
	} else {
		tmp = d * ((1.0 / 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 = sqrt((d / h))
    if (l <= (-5.1d+45)) then
        tmp = (t_0 * (sqrt(-d) * sqrt(((-1.0d0) / l)))) * (1.0d0 - ((d_1 * (d_1 * (0.125d0 * (h * (m * m))))) / (d * (d * l))))
    else if (l <= 1.52d-86) then
        tmp = (t_0 * (1.0d0 - (h * (((m * d_1) * (d_1 * (m * 0.5d0))) / ((d * 2.0d0) * (l * (d * 2.0d0))))))) / sqrt((l / d))
    else
        tmp = d * ((1.0d0 / 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 / h));
	double tmp;
	if (l <= -5.1e+45) {
		tmp = (t_0 * (Math.sqrt(-d) * Math.sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	} else if (l <= 1.52e-86) {
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / Math.sqrt((l / d));
	} else {
		tmp = d * ((1.0 / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	tmp = 0
	if l <= -5.1e+45:
		tmp = (t_0 * (math.sqrt(-d) * math.sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))))
	elif l <= 1.52e-86:
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / math.sqrt((l / d))
	else:
		tmp = d * ((1.0 / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -5.1e+45)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * Float64(1.0 - Float64(Float64(D * Float64(D * Float64(0.125 * Float64(h * Float64(M * M))))) / Float64(d * Float64(d * l)))));
	elseif (l <= 1.52e-86)
		tmp = Float64(Float64(t_0 * Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D) * Float64(D * Float64(M * 0.5))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0))))))) / sqrt(Float64(l / d)));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	tmp = 0.0;
	if (l <= -5.1e+45)
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (d * l))));
	elseif (l <= 1.52e-86)
		tmp = (t_0 * (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt((l / d));
	else
		tmp = d * ((1.0 / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.1e+45], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(D * N[(D * N[(0.125 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.52e-86], N[(N[(t$95$0 * N[(1.0 - N[(h * N[(N[(N[(M * D), $MachinePrecision] * N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.0999999999999997e45

   1. Initial program 54.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 48.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 48.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-neg.f64 62.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-\ell}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied rewrites 62.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lower-sqrt.f64 62.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   9. Applied rewrites 62.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   if -5.0999999999999997e45 < l < 1.52e-86

   1. Initial program 78.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 83.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 83.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 83.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      5. associate-/r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{1} \cdot h\right)}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\frac{1}{\frac{1}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      8. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\color{blue}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}} \cdot h\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{1 \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{1}} \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      11. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{\frac{1}{h}}}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      12. remove-double-div N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{h}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{\color{blue}{d \cdot 2}}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      18. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}}\right) \]

      19. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D}} \cdot \frac{2}{\frac{1}{2}}}\right) \]

   8. Applied rewrites 83.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d}{M \cdot D} \cdot 4}}\right) \]

   10. Applied rewrites 67.4%

       \[\leadsto \color{blue}{\frac{\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}} \]

   if 1.52e-86 < l

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 46.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 46.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 53.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 53.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      4. associate-/r* N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      5. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

      6. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \]

      9. pow1/2 N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\color{blue}{{h}^{\frac{1}{2}}}} \]

      10. metadata-eval N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      11. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      12. lower-/.f64 N/A

          \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\left(\frac{1}{2}\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      14. metadata-eval N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      15. pow1/2 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      16. sqrt-div N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      17. metadata-eval N/A

          \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{\ell}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      18. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      20. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      21. metadata-eval N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{{h}^{\color{blue}{\frac{1}{2}}}} \]

      22. pow1/2 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]

      23. lower-sqrt.f64 69.3

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]

   10. Applied rewrites 69.3%

       \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+45}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \left(d \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{t\_0}\right) \cdot \sqrt{d \cdot \frac{d}{-h}}\right) \cdot \sqrt{\frac{-1}{\ell}}\\ \mathbf{elif}\;\ell \leq -1.85 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{t\_0}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right) \cdot \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (* d l))))
   (if (<= l -1.65e+228)
     (*
      (*
       (- 1.0 (/ (* (* D 0.125) (* D (* h (* M M)))) t_0))
       (sqrt (* d (/ d (- h)))))
      (sqrt (/ -1.0 l)))
     (if (<= l -1.85e-67)
       (*
        (sqrt (/ d l))
        (*
         (sqrt (/ d h))
         (fma (- D) (/ (* (* h 0.125) (* D (* M M))) t_0) 1.0)))
       (if (<= l 6.2e-89)
         (*
          (-
           1.0
           (* h (/ (* (* M D) (* D (* M 0.5))) (* (* d 2.0) (* l (* d 2.0))))))
          (sqrt (/ (/ d h) (/ l d))))
         (* d (/ (/ 1.0 (sqrt l)) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * (d * l);
	double tmp;
	if (l <= -1.65e+228) {
		tmp = ((1.0 - (((D * 0.125) * (D * (h * (M * M)))) / t_0)) * sqrt((d * (d / -h)))) * sqrt((-1.0 / l));
	} else if (l <= -1.85e-67) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * fma(-D, (((h * 0.125) * (D * (M * M))) / t_0), 1.0));
	} else if (l <= 6.2e-89) {
		tmp = (1.0 - (h * (((M * D) * (D * (M * 0.5))) / ((d * 2.0) * (l * (d * 2.0)))))) * sqrt(((d / h) / (l / d)));
	} else {
		tmp = d * ((1.0 / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * Float64(d * l))
	tmp = 0.0
	if (l <= -1.65e+228)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(D * 0.125) * Float64(D * Float64(h * Float64(M * M)))) / t_0)) * sqrt(Float64(d * Float64(d / Float64(-h))))) * sqrt(Float64(-1.0 / l)));
	elseif (l <= -1.85e-67)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * fma(Float64(-D), Float64(Float64(Float64(h * 0.125) * Float64(D * Float64(M * M))) / t_0), 1.0)));
	elseif (l <= 6.2e-89)
		tmp = Float64(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D) * Float64(D * Float64(M * 0.5))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0)))))) * sqrt(Float64(Float64(d / h) / Float64(l / d))));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.65e+228], N[(N[(N[(1.0 - N[(N[(N[(D * 0.125), $MachinePrecision] * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d * N[(d / (-h)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.85e-67], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[((-D) * N[(N[(N[(h * 0.125), $MachinePrecision] * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e-89], N[(N[(1.0 - N[(h * N[(N[(N[(M * D), $MachinePrecision] * N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.65000000000000003e228

   1. Initial program 44.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 38.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 38.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-neg.f64 68.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-\ell}}}\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied rewrites 68.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{1}{-\ell}}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   9. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \left(-d\right)}\right) \cdot \sqrt{\frac{-1}{\ell}}} \]

   if -1.65000000000000003e228 < l < -1.85e-67

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 55.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 55.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   7. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(\left(M \cdot M\right) \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -1.85e-67 < l < 6.19999999999999993e-89

   1. Initial program 75.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied rewrites 81.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      5. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      10. lower-/.f64 82.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied rewrites 82.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}{\frac{1}{h}}\right) \]

      5. associate-/r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\left(\frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{1} \cdot h\right)}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\color{blue}{\frac{1}{\frac{1}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}} \cdot h\right)\right) \]

      8. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{1}{\color{blue}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}} \cdot h\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{1 \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{1}} \cdot h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      11. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{\frac{1}{\frac{1}{h}}}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      12. remove-double-div N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\color{blue}{h}}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d \cdot 2}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{\color{blue}{d \cdot 2}}{\frac{1}{2} \cdot \left(M \cdot D\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}}\right) \]

      18. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\frac{d \cdot 2}{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}}\right) \]

      19. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D} \cdot \frac{2}{\frac{1}{2}}}}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M \cdot D}} \cdot \frac{2}{\frac{1}{2}}}\right) \]

   8. Applied rewrites 82.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \color{blue}{\frac{h}{\frac{d}{M \cdot D} \cdot 4}}\right) \]

   10. Applied rewrites 64.1%

       \[\leadsto \color{blue}{\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right) \cdot \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \]

   if 6.19999999999999993e-89 < l

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 46.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 46.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 53.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 53.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      4. associate-/r* N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      5. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

      6. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \]

      9. pow1/2 N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\color{blue}{{h}^{\frac{1}{2}}}} \]

      10. metadata-eval N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      11. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      12. lower-/.f64 N/A

          \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\left(\frac{1}{2}\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      14. metadata-eval N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      15. pow1/2 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      16. sqrt-div N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      17. metadata-eval N/A

          \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{\ell}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      18. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      20. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      21. metadata-eval N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{{h}^{\color{blue}{\frac{1}{2}}}} \]

      22. pow1/2 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]

      23. lower-sqrt.f64 69.3

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]

   10. Applied rewrites 69.3%

       \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(1 - \frac{\left(D \cdot 0.125\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \sqrt{d \cdot \frac{d}{-h}}\right) \cdot \sqrt{\frac{-1}{\ell}}\\ \mathbf{elif}\;\ell \leq -1.85 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;\left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right) \cdot \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.35e-86)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (fma (- D) (/ (* (* h 0.125) (* D (* M M))) (* d (* d l))) 1.0)))
   (* d (/ (/ 1.0 (sqrt l)) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.35e-86) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * fma(-D, (((h * 0.125) * (D * (M * M))) / (d * (d * l))), 1.0));
	} else {
		tmp = d * ((1.0 / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.35e-86)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * fma(Float64(-D), Float64(Float64(Float64(h * 0.125) * Float64(D * Float64(M * M))) / Float64(d * Float64(d * l))), 1.0)));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.35e-86], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[((-D) * N[(N[(N[(h * 0.125), $MachinePrecision] * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.34999999999999996e-86

   1. Initial program 71.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 56.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 56.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   7. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(\left(M \cdot M\right) \cdot D\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if 1.34999999999999996e-86 < l

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 46.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 46.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 53.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 53.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      4. associate-/r* N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      5. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

      6. pow1/2 N/A

         \[\leadsto d \cdot \frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \]

      9. pow1/2 N/A

         \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\color{blue}{{h}^{\frac{1}{2}}}} \]

      10. metadata-eval N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      11. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      12. lower-/.f64 N/A

          \[\leadsto d \cdot \color{blue}{\frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{{h}^{\left(\frac{1}{2}\right)}}} \]

      13. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      14. metadata-eval N/A

          \[\leadsto d \cdot \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      15. pow1/2 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\sqrt{\frac{1}{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      16. sqrt-div N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      17. metadata-eval N/A

          \[\leadsto d \cdot \frac{\frac{\color{blue}{1}}{\sqrt{\ell}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      18. lower-/.f64 N/A

          \[\leadsto d \cdot \frac{\color{blue}{\frac{1}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\color{blue}{\sqrt{\ell}}}}{{h}^{\left(\frac{1}{2}\right)}} \]

      20. lift-/.f64 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{{h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      21. metadata-eval N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{{h}^{\color{blue}{\frac{1}{2}}}} \]

      22. pow1/2 N/A

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]

      23. lower-sqrt.f64 69.3

          \[\leadsto d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]

   10. Applied rewrites 69.3%

       \[\leadsto d \cdot \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{h \cdot \left(\ell \cdot \left(h \cdot \ell\right)\right)}}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{M \cdot M}{d} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.1e-145)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (if (<= l 2e-309)
     (/ d (sqrt (sqrt (* h (* l (* h l))))))
     (if (<= l 3.3e-168)
       (* (sqrt (/ h (* l (* l l)))) (* (* D D) (* (/ (* M M) d) -0.125)))
       (/ (/ d (sqrt h)) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.1e-145) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (l <= 2e-309) {
		tmp = d / sqrt(sqrt((h * (l * (h * l)))));
	} else if (l <= 3.3e-168) {
		tmp = sqrt((h / (l * (l * l)))) * ((D * D) * (((M * M) / d) * -0.125));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.1d-145)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else if (l <= 2d-309) then
        tmp = d / sqrt(sqrt((h * (l * (h * l)))))
    else if (l <= 3.3d-168) then
        tmp = sqrt((h / (l * (l * l)))) * ((d_1 * d_1) * (((m * m) / d) * (-0.125d0)))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.1e-145) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2e-309) {
		tmp = d / Math.sqrt(Math.sqrt((h * (l * (h * l)))));
	} else if (l <= 3.3e-168) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D * D) * (((M * M) / d) * -0.125));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.1e-145:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	elif l <= 2e-309:
		tmp = d / math.sqrt(math.sqrt((h * (l * (h * l)))))
	elif l <= 3.3e-168:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D * D) * (((M * M) / d) * -0.125))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.1e-145)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 2e-309)
		tmp = Float64(d / sqrt(sqrt(Float64(h * Float64(l * Float64(h * l))))));
	elseif (l <= 3.3e-168)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * D) * Float64(Float64(Float64(M * M) / d) * -0.125)));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.1e-145)
		tmp = -d * sqrt((1.0 / (h * l)));
	elseif (l <= 2e-309)
		tmp = d / sqrt(sqrt((h * (l * (h * l)))));
	elseif (l <= 3.3e-168)
		tmp = sqrt((h / (l * (l * l)))) * ((D * D) * (((M * M) / d) * -0.125));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.1e-145], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e-309], N[(d / N[Sqrt[N[Sqrt[N[(h * N[(l * N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.3e-168], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.1e-145

   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 M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 50.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 50.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 43.5

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.1e-145 < l < 1.9999999999999988e-309

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 62.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 16.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 16.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-sqrt.f64 16.4

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 16.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. lift-*.f64 16.4

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

       2. /-rgt-identity N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\frac{h \cdot \ell}{1}}}} \]

       3. clear-num N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\frac{1}{\frac{1}{h \cdot \ell}}}}} \]

       4. inv-pow N/A

          \[\leadsto \frac{d}{\sqrt{\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}}} \]

       5. pow-flip N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{d}{\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{1}}}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{d}{\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}}} \]

       8. sqrt-pow1 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{2}}}}} \]

       9. pow2 N/A

          \[\leadsto \frac{d}{\sqrt{\sqrt{\color{blue}{\left(h \cdot \ell\right) \cdot \left(h \cdot \ell\right)}}}} \]

       10. lower-sqrt.f64 N/A

           \[\leadsto \frac{d}{\sqrt{\color{blue}{\sqrt{\left(h \cdot \ell\right) \cdot \left(h \cdot \ell\right)}}}} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{d}{\sqrt{\sqrt{\color{blue}{\left(h \cdot \ell\right)} \cdot \left(h \cdot \ell\right)}}} \]

       12. associate-*l* N/A

           \[\leadsto \frac{d}{\sqrt{\sqrt{\color{blue}{h \cdot \left(\ell \cdot \left(h \cdot \ell\right)\right)}}}} \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{d}{\sqrt{\sqrt{\color{blue}{h \cdot \left(\ell \cdot \left(h \cdot \ell\right)\right)}}}} \]

       14. lower-*.f64 38.0

           \[\leadsto \frac{d}{\sqrt{\sqrt{h \cdot \color{blue}{\left(\ell \cdot \left(h \cdot \ell\right)\right)}}}} \]

   12. Applied rewrites 38.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{\sqrt{h \cdot \left(\ell \cdot \left(h \cdot \ell\right)\right)}}}} \]

   if 1.9999999999999988e-309 < l < 3.3000000000000001e-168

   1. Initial program 78.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 50.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 50.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left({M}^{2} \cdot {D}^{2}\right)}}{d} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot {D}^{2}}}{d} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot {M}^{2}\right)}}{d} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left({D}^{2} \cdot \frac{\frac{-1}{8} \cdot {M}^{2}}{d}\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left({D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)} \]

      18. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2}}{d}}\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      23. lower-*.f64 53.8

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(-0.125 \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

   8. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(-0.125 \cdot \frac{M \cdot M}{d}\right)\right)} \]

   if 3.3000000000000001e-168 < l

   1. Initial program 70.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 53.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 53.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 52.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 52.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-sqrt.f64 52.8

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 52.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       2. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{h}}}}{\sqrt{\ell}} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \]

       6. lower-sqrt.f64 66.7

          \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]

   12. Applied rewrites 66.7%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{h \cdot \left(\ell \cdot \left(h \cdot \ell\right)\right)}}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{M \cdot M}{d} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-175}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 3.6e-175)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (/ d (* (sqrt h) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 3.6e-175) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 3.6d-175) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 3.6e-175) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 3.6e-175:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 3.6e-175)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 3.6e-175)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 3.6e-175], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.6e-175

   1. Initial program 69.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 53.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 53.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 36.0

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 3.6e-175 < l

   1. Initial program 70.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 53.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 53.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 52.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 52.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-sqrt.f64 52.8

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 52.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       5. lower-sqrt.f64 66.6

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied rewrites 66.6%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-175}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 41.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-175}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= l 3.6e-175) (* (- d) t_0) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 3.6e-175) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (l <= 3.6d-175) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 3.6e-175) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= 3.6e-175:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= 3.6e-175)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= 3.6e-175)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 3.6e-175], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.6e-175

   1. Initial program 69.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 53.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 53.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 36.0

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 3.6e-175 < l

   1. Initial program 70.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 53.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Applied rewrites 53.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 52.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Applied rewrites 52.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-175}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 26.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt((1.0 / (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 * sqrt((1.0d0 / (h * l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt((1.0 / (h * l)))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in M around 0

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   2. lower-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   3. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

   4. unpow2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

   6. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

   7. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   9. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

   10. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

   13. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   15. unpow2 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   17. unpow2 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

   18. associate-*l* N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   20. lower-*.f64 53.4

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

5. Applied rewrites 53.4%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

6. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   3. lower-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   4. lower-*.f64 25.3

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

8. Applied rewrites 25.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
9. Add Preprocessing

Alternative 24: 26.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((h * l));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in M around 0

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   2. lower-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   3. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

   4. unpow2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

   6. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

   7. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   9. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

   10. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

   13. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   15. unpow2 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   17. unpow2 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

   18. associate-*l* N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   20. lower-*.f64 53.4

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

5. Applied rewrites 53.4%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

6. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   3. lower-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   4. lower-*.f64 25.3

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

8. Applied rewrites 25.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   2. sqrt-div N/A

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

   3. metadata-eval N/A

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   6. lower-sqrt.f64 25.3

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

10. Applied rewrites 25.3%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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)))))