Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 83.1%

Time: 9.4s

Alternatives: 21

Speedup: 3.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    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: 83.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ t_1 := \sqrt{-d}\\ t_2 := t\_0 \cdot M\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot \frac{t\_1}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D\_m}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \frac{t\_2 \cdot \left(t\_2 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot t\_0\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0)) (t_1 (sqrt (- d))) (t_2 (* t_0 M)))
   (if (<= l -4e-310)
     (*
      (* (/ t_1 (sqrt (- h))) (/ t_1 (sqrt (- l))))
      (- 1.0 (/ (* (pow (* (/ D_m 2.0) (/ M d)) 2.0) (* 0.5 h)) l)))
     (if (<= l 7.5e-134)
       (*
        (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h)))
        (- 1.0 (/ (* t_2 (* t_2 (* 0.5 h))) l)))
       (/
        (/
         (*
          (* (fma (* (pow (* M t_0) 2.0) -0.5) (/ h l) 1.0) (sqrt d))
          (sqrt d))
         (sqrt h))
        (sqrt l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double t_1 = sqrt(-d);
	double t_2 = t_0 * M;
	double tmp;
	if (l <= -4e-310) {
		tmp = ((t_1 / sqrt(-h)) * (t_1 / sqrt(-l))) * (1.0 - ((pow(((D_m / 2.0) * (M / d)), 2.0) * (0.5 * h)) / l));
	} else if (l <= 7.5e-134) {
		tmp = (sqrt((d / l)) * (sqrt(d) / sqrt(h))) * (1.0 - ((t_2 * (t_2 * (0.5 * h))) / l));
	} else {
		tmp = (((fma((pow((M * t_0), 2.0) * -0.5), (h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(D_m / d) / 2.0)
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(t_0 * M)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(t_1 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64((Float64(Float64(D_m / 2.0) * Float64(M / d)) ^ 2.0) * Float64(0.5 * h)) / l)));
	elseif (l <= 7.5e-134)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(t_2 * Float64(t_2 * Float64(0.5 * h))) / l)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64((Float64(M * t_0) ^ 2.0) * -0.5), Float64(h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * M), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e-134], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$2 * N[(t$95$2 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Power[N[(M * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 67.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 67.9

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

   3. Applied rewrites 67.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 84.2

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

   7. Applied rewrites 84.2%

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

   if -3.999999999999988e-310 < l < 7.50000000000000048e-134

   1. Initial program 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 75.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 75.0

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

   3. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.4

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 75.4

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 75.4

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

   7. Applied rewrites 75.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 86.3

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

   9. Applied rewrites 86.3%

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

   if 7.50000000000000048e-134 < l

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 65.6

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 65.6

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

   3. Applied rewrites 65.6%

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 80.2%

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

Alternative 2: 64.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\_m\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{t\_0 \cdot t\_0}{4}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) D_m)))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
        -5e-152)
     (*
      (fma -0.5 (* (/ h l) (/ (* t_0 t_0) 4.0)) 1.0)
      (sqrt (* (/ d l) (/ d h))))
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * D_m;
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-152) {
		tmp = fma(-0.5, ((h / l) * ((t_0 * t_0) / 4.0)), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-152)
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * Float64(Float64(t_0 * t_0) / 4.0)), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-152], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999997e-152

   1. Initial program 85.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.7

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

   3. Applied rewrites 74.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 74.4

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 74.4%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -4.9999999999999997e-152 < (*.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 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 59.2

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 59.2

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

   3. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.0

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 60.0

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 60.0

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 59.1%

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

Alternative 3: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(\frac{M \cdot D\_m}{d \cdot 4} \cdot \left(\frac{M}{d} \cdot D\_m\right)\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-152)
   (*
    (fma -0.5 (* (/ h l) (* (/ (* M D_m) (* d 4.0)) (* (/ M d) D_m))) 1.0)
    (sqrt (* (/ d l) (/ d h))))
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-152) {
		tmp = fma(-0.5, ((h / l) * (((M * D_m) / (d * 4.0)) * ((M / d) * D_m))), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-152)
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * Float64(Float64(Float64(M * D_m) / Float64(d * 4.0)) * Float64(Float64(M / d) * D_m))), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-152], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M * D$95$m), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999997e-152

   1. Initial program 85.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.7

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

   3. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 74.9

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 73.7

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 73.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \left(\frac{M}{d} \cdot D\right)\right)}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -4.9999999999999997e-152 < (*.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 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 59.2

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 59.2

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

   3. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.0

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 60.0

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 60.0

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 59.1%

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

Alternative 4: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(M \cdot \left(D\_m \cdot \left(\frac{M}{d \cdot 4} \cdot \frac{D\_m}{d}\right)\right)\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-152)
   (*
    (fma -0.5 (* (/ h l) (* M (* D_m (* (/ M (* d 4.0)) (/ D_m d))))) 1.0)
    (sqrt (* (/ d l) (/ d h))))
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-152) {
		tmp = fma(-0.5, ((h / l) * (M * (D_m * ((M / (d * 4.0)) * (D_m / d))))), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-152)
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * Float64(M * Float64(D_m * Float64(Float64(M / Float64(d * 4.0)) * Float64(D_m / d))))), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-152], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[(M * N[(D$95$m * N[(N[(M / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999997e-152

   1. Initial program 85.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.7

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

   3. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 74.9

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 70.0

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(M \cdot \left(D \cdot \left(\frac{M}{d \cdot \color{blue}{4}} \cdot \frac{D}{d}\right)\right)\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 70.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \color{blue}{\left(M \cdot \left(D \cdot \left(\frac{M}{d \cdot 4} \cdot \frac{D}{d}\right)\right)\right)}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -4.9999999999999997e-152 < (*.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 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 59.2

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 59.2

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

   3. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.0

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 60.0

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 60.0

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 59.1%

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

Alternative 5: 62.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(\frac{M}{d \cdot 4} \cdot \left(\left(\frac{M}{d} \cdot D\_m\right) \cdot D\_m\right)\right), 1\right) \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
      0.0)
   (*
    (fma -0.5 (* (/ h l) (* (/ M (* d 4.0)) (* (* (/ M d) D_m) D_m))) 1.0)
    (sqrt (* d (/ d (* l h)))))
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 0.0) {
		tmp = fma(-0.5, ((h / l) * ((M / (d * 4.0)) * (((M / d) * D_m) * D_m))), 1.0) * sqrt((d * (d / (l * h))));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 0.0)
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * Float64(Float64(M / Float64(d * 4.0)) * Float64(Float64(Float64(M / d) * D_m) * D_m))), 1.0) * sqrt(Float64(d * Float64(d / Float64(l * h)))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[(N[(M / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d * N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.8

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

   3. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 69.1

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 49.6

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

   7. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 63.8

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(\frac{M}{d \cdot 4} \cdot \color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)}\right), 1\right) \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}} \]

   9. Applied rewrites 63.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M}{d \cdot 4} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot D\right)\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}} \]

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

   1. Initial program 58.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 61.5

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 61.5

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.5

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 62.5

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 62.5

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

   7. Applied rewrites 62.5%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 61.4%

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

Alternative 6: 57.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{\left(M \cdot D\_m\right) \cdot \left(M \cdot D\_m\right)}{\left(d \cdot d\right) \cdot 4}, 1\right) \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-152)
   (*
    (fma -0.5 (* (/ h l) (/ (* (* M D_m) (* M D_m)) (* (* d d) 4.0))) 1.0)
    (sqrt (* d (/ d (* l h)))))
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-152) {
		tmp = fma(-0.5, ((h / l) * (((M * D_m) * (M * D_m)) / ((d * d) * 4.0))), 1.0) * sqrt((d * (d / (l * h))));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-152)
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * Float64(Float64(Float64(M * D_m) * Float64(M * D_m)) / Float64(Float64(d * d) * 4.0))), 1.0) * sqrt(Float64(d * Float64(d / Float64(l * h)))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-152], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(M * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d * N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999997e-152

   1. Initial program 85.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.7

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

   3. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 74.9

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. swap-sqr N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.5

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot 4}, 1\right) \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}} \]

   9. Applied rewrites 55.5%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot d\right) \cdot 4}}, 1\right) \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}} \]

   if -4.9999999999999997e-152 < (*.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 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 59.2

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 59.2

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

   3. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.0

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 60.0

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 60.0

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 59.1%

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

Alternative 7: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ t_1 := t\_0 \cdot M\\ t_2 := 1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_3 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;\left(t\_3 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot t\_0\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0))
        (t_1 (* t_0 M))
        (t_2 (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l)))
        (t_3 (sqrt (/ d l))))
   (if (<= l -4e-310)
     (* (* t_3 (/ (sqrt (- d)) (sqrt (- h)))) t_2)
     (if (<= l 7.5e-134)
       (* (* t_3 (/ (sqrt d) (sqrt h))) t_2)
       (/
        (/
         (*
          (* (fma (* (pow (* M t_0) 2.0) -0.5) (/ h l) 1.0) (sqrt d))
          (sqrt d))
         (sqrt h))
        (sqrt l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double t_1 = t_0 * M;
	double t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	double t_3 = sqrt((d / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_3 * (sqrt(-d) / sqrt(-h))) * t_2;
	} else if (l <= 7.5e-134) {
		tmp = (t_3 * (sqrt(d) / sqrt(h))) * t_2;
	} else {
		tmp = (((fma((pow((M * t_0), 2.0) * -0.5), (h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(D_m / d) / 2.0)
	t_1 = Float64(t_0 * M)
	t_2 = Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_3 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * t_2);
	elseif (l <= 7.5e-134)
		tmp = Float64(Float64(t_3 * Float64(sqrt(d) / sqrt(h))) * t_2);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64((Float64(M * t_0) ^ 2.0) * -0.5), Float64(h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * M), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$3 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, 7.5e-134], N[(N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Power[N[(M * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 67.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 67.9

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

   3. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 69.0

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 69.0

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

   7. Applied rewrites 69.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 79.0

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

   9. Applied rewrites 79.0%

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

   if -3.999999999999988e-310 < l < 7.50000000000000048e-134

   1. Initial program 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 75.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 75.0

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

   3. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.4

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 75.4

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 75.4

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

   7. Applied rewrites 75.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 86.3

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

   9. Applied rewrites 86.3%

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

   if 7.50000000000000048e-134 < l

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 65.6

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 65.6

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

   3. Applied rewrites 65.6%

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 80.2%

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

Alternative 8: 81.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{{\left(\frac{D\_m}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M)))
   (if (<= l -4e-310)
     (*
      (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h))))
      (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
     (*
      (/ (* (sqrt d) (sqrt d)) (* (sqrt l) (sqrt h)))
      (- 1.0 (/ (* (pow (* (/ D_m 2.0) (/ M d)) 2.0) (* 0.5 h)) l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double tmp;
	if (l <= -4e-310) {
		tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else {
		tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * (1.0 - ((pow(((D_m / 2.0) * (M / d)), 2.0) * (0.5 * h)) / l));
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_m)
use fmin_fmax_functions
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d_m / d) / 2.0d0) * m
    if (l <= (-4d-310)) then
        tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * (1.0d0 - ((t_0 * (t_0 * (0.5d0 * h))) / l))
    else
        tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * (1.0d0 - (((((d_m / 2.0d0) * (m / d)) ** 2.0d0) * (0.5d0 * h)) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double tmp;
	if (l <= -4e-310) {
		tmp = (Math.sqrt((d / l)) * (Math.sqrt(-d) / Math.sqrt(-h))) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else {
		tmp = ((Math.sqrt(d) * Math.sqrt(d)) / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((Math.pow(((D_m / 2.0) * (M / d)), 2.0) * (0.5 * h)) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = ((D_m / d) / 2.0) * M
	tmp = 0
	if l <= -4e-310:
		tmp = (math.sqrt((d / l)) * (math.sqrt(-d) / math.sqrt(-h))) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l))
	else:
		tmp = ((math.sqrt(d) * math.sqrt(d)) / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((math.pow(((D_m / 2.0) * (M / d)), 2.0) * (0.5 * h)) / l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(d)) / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64((Float64(Float64(D_m / 2.0) * Float64(M / d)) ^ 2.0) * Float64(0.5 * h)) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = ((D_m / d) / 2.0) * M;
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	else
		tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * (1.0 - (((((D_m / 2.0) * (M / d)) ^ 2.0) * (0.5 * h)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 67.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 67.9

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

   3. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 69.0

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 69.0

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

   7. Applied rewrites 69.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 79.0

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

   9. Applied rewrites 79.0%

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

   if -3.999999999999988e-310 < l

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 68.2

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 68.2

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

   3. Applied rewrites 68.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. sqrt-div N/A

          \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d}} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      20. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      22. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      23. lower-sqrt.f64 84.4

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+63}:\\ \;\;\;\;\frac{t\_2 \cdot \sqrt{d}}{\sqrt{h}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(\left(M \cdot D\_m\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D\_m}{4}}{d} \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
        (t_2 (sqrt (/ d l))))
   (if (<= l -4e-310)
     (* (* t_2 (/ (sqrt (- d)) (sqrt (- h)))) t_1)
     (if (<= l 1.65e+63)
       (* (/ (* t_2 (sqrt d)) (sqrt h)) t_1)
       (/
        (/
         (*
          (*
           (fma
            (* (* M D_m) (* (/ (/ (* (/ M d) D_m) 4.0) d) -0.5))
            (/ h l)
            1.0)
           (sqrt d))
          (sqrt d))
         (sqrt h))
        (sqrt l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_2 * (sqrt(-d) / sqrt(-h))) * t_1;
	} else if (l <= 1.65e+63) {
		tmp = ((t_2 * sqrt(d)) / sqrt(h)) * t_1;
	} else {
		tmp = (((fma(((M * D_m) * (((((M / d) * D_m) / 4.0) / d) * -0.5)), (h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_2 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * t_1);
	elseif (l <= 1.65e+63)
		tmp = Float64(Float64(Float64(t_2 * sqrt(d)) / sqrt(h)) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(M * D_m) * Float64(Float64(Float64(Float64(Float64(M / d) * D_m) / 4.0) / d) * -0.5)), Float64(h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$2 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.65e+63], N[(N[(N[(t$95$2 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision] / 4.0), $MachinePrecision] / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 66.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 67.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 67.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 67.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 69.0

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 69.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 69.0

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. lower-neg.f64 79.0

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 79.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if -3.999999999999988e-310 < l < 1.6500000000000001e63

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 75.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 75.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 75.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 76.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d}}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d}}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d}}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. unpow1/2 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      18. lower-sqrt.f64 86.5

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 1.6500000000000001e63 < l

   1. Initial program 57.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 56.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 56.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 56.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

   7. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      15. times-frac N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

   9. Applied rewrites 73.0%

      \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D}{4}}{d} \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_2 \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+63}:\\ \;\;\;\;\frac{t\_2 \cdot \sqrt{d}}{\sqrt{h}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(\left(M \cdot D\_m\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D\_m}{4}}{d} \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
        (t_2 (sqrt (/ d l))))
   (if (<= l -4e-310)
     (* (* t_2 (sqrt (/ d h))) t_1)
     (if (<= l 1.65e+63)
       (* (/ (* t_2 (sqrt d)) (sqrt h)) t_1)
       (/
        (/
         (*
          (*
           (fma
            (* (* M D_m) (* (/ (/ (* (/ M d) D_m) 4.0) d) -0.5))
            (/ h l)
            1.0)
           (sqrt d))
          (sqrt d))
         (sqrt h))
        (sqrt l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_2 * sqrt((d / h))) * t_1;
	} else if (l <= 1.65e+63) {
		tmp = ((t_2 * sqrt(d)) / sqrt(h)) * t_1;
	} else {
		tmp = (((fma(((M * D_m) * (((((M / d) * D_m) / 4.0) / d) * -0.5)), (h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_2 * sqrt(Float64(d / h))) * t_1);
	elseif (l <= 1.65e+63)
		tmp = Float64(Float64(Float64(t_2 * sqrt(d)) / sqrt(h)) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(M * D_m) * Float64(Float64(Float64(Float64(Float64(M / d) * D_m) / 4.0) / d) * -0.5)), Float64(h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.65e+63], N[(N[(N[(t$95$2 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision] / 4.0), $MachinePrecision] / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 66.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 67.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 67.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 67.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 69.0

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 69.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 69.0

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if -3.999999999999988e-310 < l < 1.6500000000000001e63

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 75.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 75.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 75.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 76.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d}}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d}}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d}}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. unpow1/2 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{d}}{\sqrt{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      18. lower-sqrt.f64 86.5

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}}{\sqrt{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 1.6500000000000001e63 < l

   1. Initial program 57.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 56.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 56.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 56.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

   7. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      15. times-frac N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

   9. Applied rewrites 73.0%

      \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D}{4}}{d} \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_2 \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+19}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(\left(M \cdot D\_m\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D\_m}{4}}{d} \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
        (t_2 (sqrt (/ d l))))
   (if (<= l -4e-310)
     (* (* t_2 (sqrt (/ d h))) t_1)
     (if (<= l 8.2e+19)
       (* (* t_2 (/ (sqrt d) (sqrt h))) t_1)
       (/
        (/
         (*
          (*
           (fma
            (* (* M D_m) (* (/ (/ (* (/ M d) D_m) 4.0) d) -0.5))
            (/ h l)
            1.0)
           (sqrt d))
          (sqrt d))
         (sqrt h))
        (sqrt l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_2 * sqrt((d / h))) * t_1;
	} else if (l <= 8.2e+19) {
		tmp = (t_2 * (sqrt(d) / sqrt(h))) * t_1;
	} else {
		tmp = (((fma(((M * D_m) * (((((M / d) * D_m) / 4.0) / d) * -0.5)), (h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_2 * sqrt(Float64(d / h))) * t_1);
	elseif (l <= 8.2e+19)
		tmp = Float64(Float64(t_2 * Float64(sqrt(d) / sqrt(h))) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(M * D_m) * Float64(Float64(Float64(Float64(Float64(M / d) * D_m) / 4.0) / d) * -0.5)), Float64(h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 8.2e+19], N[(N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision] / 4.0), $MachinePrecision] / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 66.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 67.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 67.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 67.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 69.0

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 69.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 69.0

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if -3.999999999999988e-310 < l < 8.2e19

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 76.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 76.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 76.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 77.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 77.0

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 77.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 77.0

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lower-/.f64 86.9

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 86.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 8.2e19 < l

   1. Initial program 60.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 58.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 58.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 58.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

   7. Applied rewrites 75.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      15. times-frac N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

   9. Applied rewrites 74.9%

      \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D}{4}}{d} \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\\ \mathbf{if}\;\ell \leq 1650000000000:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(\left(M \cdot D\_m\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D\_m}{4}}{d} \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M)))
   (if (<= l 1650000000000.0)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
     (/
      (/
       (*
        (*
         (fma (* (* M D_m) (* (/ (/ (* (/ M d) D_m) 4.0) d) -0.5)) (/ h l) 1.0)
         (sqrt d))
        (sqrt d))
       (sqrt h))
      (sqrt l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double tmp;
	if (l <= 1650000000000.0) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else {
		tmp = (((fma(((M * D_m) * (((((M / d) * D_m) / 4.0) / d) * -0.5)), (h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M)
	tmp = 0.0
	if (l <= 1650000000000.0)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(M * D_m) * Float64(Float64(Float64(Float64(Float64(M / d) * D_m) / 4.0) / d) * -0.5)), Float64(h / l), 1.0) * sqrt(d)) * sqrt(d)) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[l, 1650000000000.0], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision] / 4.0), $MachinePrecision] / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.65e12

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 70.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 70.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 70.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 71.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 71.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 71.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 71.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 1.65e12 < l

   1. Initial program 60.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 59.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 59.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 59.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

   7. Applied rewrites 76.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      15. times-frac N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]

   9. Applied rewrites 75.2%

      \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{M}{d} \cdot D}{4}}{d} \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}{\sqrt{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 70.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\_m\\ t_1 := \frac{\frac{D\_m}{d}}{2}\\ t_2 := t\_1 \cdot M\\ \mathbf{if}\;d \leq -1.14 \cdot 10^{+216}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-80}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - t\_1 \cdot \left(M \cdot \left(\left(0.5 \cdot \left(M \cdot t\_1\right)\right) \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{\left(h \cdot t\_0\right) \cdot \frac{t\_0}{4}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{t\_2 \cdot \left(t\_2 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) D_m)) (t_1 (/ (/ D_m d) 2.0)) (t_2 (* t_1 M)))
   (if (<= d -1.14e+216)
     (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
     (if (<= d -1.15e-80)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (- 1.0 (* t_1 (* M (* (* 0.5 (* M t_1)) (/ h l))))))
       (if (<= d 5e-304)
         (*
          (fma -0.5 (/ (* (* h t_0) (/ t_0 4.0)) l) 1.0)
          (sqrt (* (/ d l) (/ d h))))
         (* (/ d (sqrt (* h l))) (- 1.0 (/ (* t_2 (* t_2 (* 0.5 h))) l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * D_m;
	double t_1 = (D_m / d) / 2.0;
	double t_2 = t_1 * M;
	double tmp;
	if (d <= -1.14e+216) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (d <= -1.15e-80) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_1 * (M * ((0.5 * (M * t_1)) * (h / l)))));
	} else if (d <= 5e-304) {
		tmp = fma(-0.5, (((h * t_0) * (t_0 / 4.0)) / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - ((t_2 * (t_2 * (0.5 * h))) / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * D_m)
	t_1 = Float64(Float64(D_m / d) / 2.0)
	t_2 = Float64(t_1 * M)
	tmp = 0.0
	if (d <= -1.14e+216)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= -1.15e-80)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(t_1 * Float64(M * Float64(Float64(0.5 * Float64(M * t_1)) * Float64(h / l))))));
	elseif (d <= 5e-304)
		tmp = Float64(fma(-0.5, Float64(Float64(Float64(h * t_0) * Float64(t_0 / 4.0)) / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(t_2 * Float64(t_2 * Float64(0.5 * h))) / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * M), $MachinePrecision]}, If[LessEqual[d, -1.14e+216], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.15e-80], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[(M * N[(N[(0.5 * N[(M * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e-304], N[(N[(-0.5 * N[(N[(N[(h * t$95$0), $MachinePrecision] * N[(t$95$0 / 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$2 * N[(t$95$2 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.14e216

   1. Initial program 70.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 70.6

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 64.4

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 64.4%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 63.5

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 63.5%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 72.8

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   10. Applied rewrites 72.8%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if -1.14e216 < d < -1.1499999999999999e-80

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 79.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 79.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 79.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 81.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 81.4

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 81.4

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 81.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)} \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{D}{d}}{2} \cdot \color{blue}{\left(M \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right)}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{\color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \frac{1}{2}\right) \cdot h}}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{D}{d}}{2} \cdot \left(M \cdot \color{blue}{\left(\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{D}{d}}{2} \cdot \left(M \cdot \left(\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right)\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{D}{d}}{2} \cdot \left(M \cdot \color{blue}{\left(\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

   9. Applied rewrites 78.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{\frac{D}{d}}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

   if -1.1499999999999999e-80 < d < 4.99999999999999965e-304

   1. Initial program 48.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 48.6

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 39.3

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 38.5

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 38.5%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 39.8

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 39.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \color{blue}{\left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(\left(2 \cdot d\right) \cdot 2\right) \cdot d}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot 2} \cdot \frac{M \cdot D}{d}\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   9. Applied rewrites 40.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(h \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot D}{4}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if 4.99999999999999965e-304 < d

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 68.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 68.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 68.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 69.5

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 69.5

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 69.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lower-sqrt.f64 76.2

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. lower-*.f64 76.2

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 70.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\_m\\ t_1 := \frac{\frac{D\_m}{d}}{2}\\ t_2 := t\_1 \cdot M\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+215}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-80}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(M \cdot D\_m\right) \cdot \left(\left(0.5 \cdot \left(M \cdot t\_1\right)\right) \cdot \frac{h}{\ell}\right)}{2 \cdot d}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{\left(h \cdot t\_0\right) \cdot \frac{t\_0}{4}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{t\_2 \cdot \left(t\_2 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) D_m)) (t_1 (/ (/ D_m d) 2.0)) (t_2 (* t_1 M)))
   (if (<= d -9.5e+215)
     (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
     (if (<= d -1.15e-80)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (- 1.0 (/ (* (* M D_m) (* (* 0.5 (* M t_1)) (/ h l))) (* 2.0 d))))
       (if (<= d 5e-304)
         (*
          (fma -0.5 (/ (* (* h t_0) (/ t_0 4.0)) l) 1.0)
          (sqrt (* (/ d l) (/ d h))))
         (* (/ d (sqrt (* h l))) (- 1.0 (/ (* t_2 (* t_2 (* 0.5 h))) l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * D_m;
	double t_1 = (D_m / d) / 2.0;
	double t_2 = t_1 * M;
	double tmp;
	if (d <= -9.5e+215) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (d <= -1.15e-80) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((M * D_m) * ((0.5 * (M * t_1)) * (h / l))) / (2.0 * d)));
	} else if (d <= 5e-304) {
		tmp = fma(-0.5, (((h * t_0) * (t_0 / 4.0)) / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - ((t_2 * (t_2 * (0.5 * h))) / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * D_m)
	t_1 = Float64(Float64(D_m / d) / 2.0)
	t_2 = Float64(t_1 * M)
	tmp = 0.0
	if (d <= -9.5e+215)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= -1.15e-80)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(M * D_m) * Float64(Float64(0.5 * Float64(M * t_1)) * Float64(h / l))) / Float64(2.0 * d))));
	elseif (d <= 5e-304)
		tmp = Float64(fma(-0.5, Float64(Float64(Float64(h * t_0) * Float64(t_0 / 4.0)) / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(t_2 * Float64(t_2 * Float64(0.5 * h))) / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * M), $MachinePrecision]}, If[LessEqual[d, -9.5e+215], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.15e-80], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(0.5 * N[(M * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e-304], N[(N[(-0.5 * N[(N[(N[(h * t$95$0), $MachinePrecision] * N[(t$95$0 / 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$2 * N[(t$95$2 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.50000000000000036e215

   1. Initial program 70.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 70.6

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 64.4

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 64.4%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 63.5

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 63.5%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 72.8

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   10. Applied rewrites 72.8%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if -9.50000000000000036e215 < d < -1.1499999999999999e-80

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 79.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 79.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 79.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 81.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 81.4

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 81.4

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 81.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)} \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)} \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. associate-*l/ N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}}{2 \cdot d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}}{2 \cdot d}}\right) \]

   9. Applied rewrites 78.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{\frac{D}{d}}{2}\right)\right) \cdot \frac{h}{\ell}\right)}{2 \cdot d}}\right) \]

   if -1.1499999999999999e-80 < d < 4.99999999999999965e-304

   1. Initial program 48.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 48.6

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 39.3

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 38.5

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 38.5%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 39.8

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 39.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \color{blue}{\left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(\left(2 \cdot d\right) \cdot 2\right) \cdot d}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot 2} \cdot \frac{M \cdot D}{d}\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   9. Applied rewrites 40.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(h \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot D}{4}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if 4.99999999999999965e-304 < d

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 68.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 68.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 68.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 69.5

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 69.5

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 69.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lower-sqrt.f64 76.2

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. lower-*.f64 76.2

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 70.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;d \leq 1.1 \cdot 10^{-214}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l))))
   (if (<= d 1.1e-214)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)
     (* (/ d (sqrt (* h l))) t_1))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double tmp;
	if (d <= 1.1e-214) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1;
	} else {
		tmp = (d / sqrt((h * l))) * t_1;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_m)
use fmin_fmax_functions
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_m / d) / 2.0d0) * m
    t_1 = 1.0d0 - ((t_0 * (t_0 * (0.5d0 * h))) / l)
    if (d <= 1.1d-214) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1
    else
        tmp = (d / sqrt((h * l))) * t_1
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double tmp;
	if (d <= 1.1e-214) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * t_1;
	} else {
		tmp = (d / Math.sqrt((h * l))) * t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = ((D_m / d) / 2.0) * M
	t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l)
	tmp = 0
	if d <= 1.1e-214:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * t_1
	else:
		tmp = (d / math.sqrt((h * l))) * t_1
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	tmp = 0.0
	if (d <= 1.1e-214)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_1);
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * t_1);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = ((D_m / d) / 2.0) * M;
	t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	tmp = 0.0;
	if (d <= 1.1e-214)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1;
	else
		tmp = (d / sqrt((h * l))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 1.1e-214], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.10000000000000001e-214

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 64.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 64.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 64.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 65.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 65.0

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 65.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 65.0

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 1.10000000000000001e-214 < d

   1. Initial program 72.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 73.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 73.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 73.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 74.6

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 74.6

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 74.6

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lower-sqrt.f64 78.9

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. lower-*.f64 78.9

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\_m\\ t_1 := \frac{\frac{D\_m}{d}}{2} \cdot M\\ \mathbf{if}\;d \leq -8 \cdot 10^{+215}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{\left(h \cdot t\_0\right) \cdot \frac{t\_0}{4}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) D_m)) (t_1 (* (/ (/ D_m d) 2.0) M)))
   (if (<= d -8e+215)
     (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
     (if (<= d 5e-304)
       (*
        (fma -0.5 (/ (* (* h t_0) (/ t_0 4.0)) l) 1.0)
        (sqrt (* (/ d l) (/ d h))))
       (* (/ d (sqrt (* h l))) (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * D_m;
	double t_1 = ((D_m / d) / 2.0) * M;
	double tmp;
	if (d <= -8e+215) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (d <= 5e-304) {
		tmp = fma(-0.5, (((h * t_0) * (t_0 / 4.0)) / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - ((t_1 * (t_1 * (0.5 * h))) / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * D_m)
	t_1 = Float64(Float64(Float64(D_m / d) / 2.0) * M)
	tmp = 0.0
	if (d <= -8e+215)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= 5e-304)
		tmp = Float64(fma(-0.5, Float64(Float64(Float64(h * t_0) * Float64(t_0 / 4.0)) / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -8e+215], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e-304], N[(N[(-0.5 * N[(N[(N[(h * t$95$0), $MachinePrecision] * N[(t$95$0 / 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.99999999999999925e215

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 70.5

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 63.7

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 64.3

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 64.3%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 63.4

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 63.4%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 72.8

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   10. Applied rewrites 72.8%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if -7.99999999999999925e215 < d < 4.99999999999999965e-304

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 65.3

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 56.4

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 56.2

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 56.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 56.9

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 56.9%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \color{blue}{\left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(\left(2 \cdot d\right) \cdot 2\right) \cdot d}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot 2} \cdot \frac{M \cdot D}{d}\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   9. Applied rewrites 57.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(h \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot D}{4}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if 4.99999999999999965e-304 < d

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 68.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 68.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 68.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-*.f64 69.5

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lower-sqrt.f64 69.5

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lift-sqrt.f64 69.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   7. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lower-sqrt.f64 76.2

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. lower-*.f64 76.2

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 63.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-199}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{\left(h \cdot t\_0\right) \cdot \frac{t\_0}{4}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{t\_0 \cdot t\_0}{4}, 1\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) D_m)))
   (if (<= l 1.95e-199)
     (*
      (fma -0.5 (/ (* (* h t_0) (/ t_0 4.0)) l) 1.0)
      (sqrt (* (/ d l) (/ d h))))
     (*
      (fma -0.5 (* (/ h l) (/ (* t_0 t_0) 4.0)) 1.0)
      (* d (sqrt (/ 1.0 (* h l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * D_m;
	double tmp;
	if (l <= 1.95e-199) {
		tmp = fma(-0.5, (((h * t_0) * (t_0 / 4.0)) / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = fma(-0.5, ((h / l) * ((t_0 * t_0) / 4.0)), 1.0) * (d * sqrt((1.0 / (h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * D_m)
	tmp = 0.0
	if (l <= 1.95e-199)
		tmp = Float64(fma(-0.5, Float64(Float64(Float64(h * t_0) * Float64(t_0 / 4.0)) / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * Float64(Float64(t_0 * t_0) / 4.0)), 1.0) * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, 1.95e-199], N[(N[(-0.5 * N[(N[(N[(h * t$95$0), $MachinePrecision] * N[(t$95$0 / 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9500000000000001e-199

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 60.0

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 59.8

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 59.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 60.4

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 60.4%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\color{blue}{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \color{blue}{\left(2 \cdot d\right)}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(\left(2 \cdot d\right) \cdot 2\right) \cdot d}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \color{blue}{\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot 2} \cdot \frac{M \cdot D}{d}\right)}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   9. Applied rewrites 61.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(h \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot D}{4}}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if 1.9500000000000001e-199 < l

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 66.6

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right), 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. metadata-eval 53.6

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, 1\right) \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, 1\right) \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, 1\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, 1\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-*.f64 68.2

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, 1\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, 1\right) \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-278}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l 7.5e-278)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (/ (* d (sqrt (/ 1.0 h))) (sqrt l))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 7.5e-278) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * sqrt((1.0 / h))) / sqrt(l);
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_m)
use fmin_fmax_functions
    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_m
    real(8) :: tmp
    if (l <= 7.5d-278) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d * sqrt((1.0d0 / h))) / sqrt(l)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 7.5e-278) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * Math.sqrt((1.0 / h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= 7.5e-278:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d * math.sqrt((1.0 / h))) / math.sqrt(l)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= 7.5e-278)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / h))) / sqrt(l));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= 7.5e-278)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d * sqrt((1.0 / h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, 7.5e-278], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.49999999999999946e-278

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 66.3

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 58.7

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 58.6

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 58.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 59.0

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 41.8

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   10. Applied rewrites 41.8%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 7.49999999999999946e-278 < l

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 68.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 68.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]

   8. Taylor expanded in d around inf

      \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

      3. lower-/.f64 48.4

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

   10. Applied rewrites 48.4%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-278}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l 7.5e-278)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (/ (/ d (sqrt h)) (sqrt l))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 7.5e-278) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_m)
use fmin_fmax_functions
    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_m
    real(8) :: tmp
    if (l <= 7.5d-278) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 7.5e-278) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= 7.5e-278:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= 7.5e-278)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= 7.5e-278)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, 7.5e-278], N[(-1.0 * 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 2 regimes

2. if l < 7.49999999999999946e-278

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 66.3

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 58.7

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 58.6

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 58.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 59.0

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 41.8

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   10. Applied rewrites 41.8%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 7.49999999999999946e-278 < l

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 68.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 68.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]

   8. Taylor expanded in d around inf

      \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{h}}}{\sqrt{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 48.4%

       \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{h}}}{\sqrt{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 41.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-226}:\\ \;\;\;\;-1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ 1.0 (* h l))))))
   (if (<= l 4.1e-226) (* -1.0 t_0) t_0)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 4.1e-226) {
		tmp = -1.0 * t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_m)
use fmin_fmax_functions
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * sqrt((1.0d0 / (h * l)))
    if (l <= 4.1d-226) then
        tmp = (-1.0d0) * t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 4.1e-226) {
		tmp = -1.0 * t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d * math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= 4.1e-226:
		tmp = -1.0 * t_0
	else:
		tmp = t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
	tmp = 0.0
	if (l <= 4.1e-226)
		tmp = Float64(-1.0 * t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d * sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= 4.1e-226)
		tmp = -1.0 * t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.1e-226], N[(-1.0 * t$95$0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if l < 4.10000000000000037e-226

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 66.8

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   3. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell}} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-*.f64 59.7

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot h}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D}{\color{blue}{d \cdot 2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-/.f64 59.5

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\left(\frac{\color{blue}{\frac{D}{d}}}{2} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   5. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2} \cdot h}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{\frac{D}{d}}{2}} \cdot M\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{\frac{D}{d} \cdot M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-/.f64 60.0

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Applied rewrites 60.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 40.5

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   10. Applied rewrites 40.5%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 4.10000000000000037e-226 < l

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 67.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 67.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

   7. Applied rewrites 80.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]

   8. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 43.8

         \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   10. Applied rewrites 43.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 25.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d * sqrt((1.0 / (h * l)));
}

Fortran

D_m =     private
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_m)
use fmin_fmax_functions
    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_m
    code = d * sqrt((1.0d0 / (h * l)))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d * math.sqrt((1.0 / (h * l)))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

   6. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

   7. associate-*l* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   11. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   13. times-frac N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

   17. lower-*.f64 68.1

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

   18. lift-/.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

   19. metadata-eval 68.1

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

3. Applied rewrites 68.1%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

5. Applied rewrites 35.6%

   \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right)}{\sqrt{\ell}}} \]

7. Applied rewrites 40.5%

   \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{h}}}}{\sqrt{\ell}} \]

8. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   3. lower-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   4. lower-*.f64 25.6

      \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

10. Applied rewrites 25.6%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025108 
(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)))))