Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 76.3%

Time: 10.9s

Alternatives: 17

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 66.7% 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: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.1 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-305}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D\_m \cdot -0.5\right)\right) \cdot \frac{M\_m}{d}\right) \cdot \left(\left(\left(M\_m \cdot \frac{-0.5}{d}\right) \cdot D\_m\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5.1e+120)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (if (<= d -2.15e-305)
     (*
      (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h))))
      (-
       1.0
       (*
        (* (* 0.5 (* D_m -0.5)) (/ M_m d))
        (* (* (* M_m (/ -0.5 d)) D_m) (/ h l)))))
     (*
      (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
      (-
       1.0
       (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (d <= -2.15e-305) {
		tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5.1d+120)) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else if (d <= (-2.15d-305)) then
        tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * (1.0d0 - (((0.5d0 * (d_m * (-0.5d0))) * (m_m / d)) * (((m_m * ((-0.5d0) / d)) * d_m) * (h / l))))
    else
        tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else if (d <= -2.15e-305) {
		tmp = (Math.sqrt((d / l)) * (Math.sqrt(-d) / Math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5.1e+120:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	elif d <= -2.15e-305:
		tmp = (math.sqrt((d / l)) * (math.sqrt(-d) / math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * (math.sqrt(d) / math.sqrt(l))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5.1e+120)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= -2.15e-305)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(D_m * -0.5)) * Float64(M_m / d)) * Float64(Float64(Float64(M_m * Float64(-0.5 / d)) * D_m) * Float64(h / l)))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5.1e+120)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	elseif (d <= -2.15e-305)
		tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	else
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5.1e+120], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.15e-305], 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[(N[(0.5 * N[(D$95$m * -0.5), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.10000000000000027e120

   1. Initial program 66.7%

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

   3. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

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

   8. Applied rewrites 25.5%

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

   if -5.10000000000000027e120 < d < -2.1500000000000001e-305

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-neg.f64 38.6

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

   9. Applied rewrites 38.6%

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

   if -2.1500000000000001e-305 < d

   1. Initial program 66.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-unsound-sqrt.f64 36.0

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

   3. Applied rewrites 36.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 40.6

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

   5. Applied rewrites 40.6%

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

Alternative 2: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D\_m \cdot -0.5\right)\right) \cdot \frac{M\_m}{d}\right) \cdot \left(\left(\left(M\_m \cdot \frac{-0.5}{d}\right) \cdot D\_m\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 1.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(-\sqrt{d}\right)}{-\sqrt{\ell}} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.3 \cdot 10^{+219}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \frac{\ell - \left(\left(\frac{\left(D\_m \cdot D\_m\right) \cdot M\_m}{d \cdot d} \cdot 0.125\right) \cdot M\_m\right) \cdot h}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(\frac{D\_m}{d + d} \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{0.5}{d}\right)\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= h -5e-310)
     (*
      (* t_0 (/ (sqrt (- d)) (sqrt (- h))))
      (-
       1.0
       (*
        (* (* 0.5 (* D_m -0.5)) (/ M_m d))
        (* (* (* M_m (/ -0.5 d)) D_m) (/ h l)))))
     (if (<= h 1.7e+113)
       (*
        (/ (* (sqrt (/ d h)) (- (sqrt d))) (- (sqrt l)))
        (- 1.0 (* (* 0.5 (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
       (if (<= h 2.3e+219)
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (/ (- l (* (* (* (/ (* (* D_m D_m) M_m) (* d d)) 0.125) M_m) h)) l))
         (*
          (* t_0 (/ (sqrt d) (sqrt h)))
          (-
           1.0
           (*
            (*
             (/ 1.0 2.0)
             (* (* (* (/ D_m (+ d d)) M_m) (* D_m M_m)) (/ 0.5 d)))
            (/ h l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	} else if (h <= 1.7e+113) {
		tmp = ((sqrt((d / h)) * -sqrt(d)) / -sqrt(l)) * (1.0 - ((0.5 * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (h <= 2.3e+219) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l);
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (h <= (-5d-310)) then
        tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0d0 - (((0.5d0 * (d_m * (-0.5d0))) * (m_m / d)) * (((m_m * ((-0.5d0) / d)) * d_m) * (h / l))))
    else if (h <= 1.7d+113) then
        tmp = ((sqrt((d / h)) * -sqrt(d)) / -sqrt(l)) * (1.0d0 - ((0.5d0 * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    else if (h <= 2.3d+219) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((l - ((((((d_m * d_m) * m_m) / (d * d)) * 0.125d0) * m_m) * h)) / l)
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((((d_m / (d + d)) * m_m) * (d_m * m_m)) * (0.5d0 / d))) * (h / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (t_0 * (Math.sqrt(-d) / Math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	} else if (h <= 1.7e+113) {
		tmp = ((Math.sqrt((d / h)) * -Math.sqrt(d)) / -Math.sqrt(l)) * (1.0 - ((0.5 * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (h <= 2.3e+219) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l);
	} else {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if h <= -5e-310:
		tmp = (t_0 * (math.sqrt(-d) / math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))))
	elif h <= 1.7e+113:
		tmp = ((math.sqrt((d / h)) * -math.sqrt(d)) / -math.sqrt(l)) * (1.0 - ((0.5 * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	elif h <= 2.3e+219:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l)
	else:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(D_m * -0.5)) * Float64(M_m / d)) * Float64(Float64(Float64(M_m * Float64(-0.5 / d)) * D_m) * Float64(h / l)))));
	elseif (h <= 1.7e+113)
		tmp = Float64(Float64(Float64(sqrt(Float64(d / h)) * Float64(-sqrt(d))) / Float64(-sqrt(l))) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	elseif (h <= 2.3e+219)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(l - Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * M_m) / Float64(d * d)) * 0.125) * M_m) * h)) / l));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(Float64(Float64(Float64(D_m / Float64(d + d)) * M_m) * Float64(D_m * M_m)) * Float64(0.5 / d))) * Float64(h / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	elseif (h <= 1.7e+113)
		tmp = ((sqrt((d / h)) * -sqrt(d)) / -sqrt(l)) * (1.0 - ((0.5 * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	elseif (h <= 2.3e+219)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l);
	else
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.5 * N[(D$95$m * -0.5), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.7e+113], N[(N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[d], $MachinePrecision])), $MachinePrecision] / (-N[Sqrt[l], $MachinePrecision])), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.3e+219], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(l - N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-neg.f64 38.6

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

   9. Applied rewrites 38.6%

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

   if -4.999999999999985e-310 < h < 1.70000000000000009e113

   1. Initial program 66.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lift-/.f64 N/A

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

      16. sqrt-undiv N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-2neg N/A

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

      20. associate-*r/ N/A

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

      21. lower-/.f64 N/A

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

   5. Applied rewrites 36.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 36.0

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

   7. Applied rewrites 36.0%

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

   if 1.70000000000000009e113 < h < 2.3000000000000001e219

   1. Initial program 66.7%

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

   3. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 27.8

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

   8. Applied rewrites 27.8%

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

   if 2.3000000000000001e219 < h

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/r* N/A

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

      22. lift-/.f64 N/A

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

      23. lower-/.f64 65.0

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

      24. lift-/.f64 N/A

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

      25. metadata-eval 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-unsound-sqrt.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-sqrt.f64 37.3

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

   7. Applied rewrites 37.3%

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

Alternative 3: 73.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(\left(M\_m \cdot \frac{-0.5}{d}\right) \cdot D\_m\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;d \leq -5.1 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-305}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D\_m \cdot -0.5\right)\right) \cdot \frac{M\_m}{d}\right) \cdot t\_1\right)\\ \mathbf{elif}\;d \leq 6.9 \cdot 10^{-193}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h \cdot \ell}} \cdot \sqrt{d}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(\frac{D\_m}{d + d} \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{0.5}{d}\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{M\_m}{d} \cdot \left(-0.25 \cdot D\_m\right)\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* (* (* M_m (/ -0.5 d)) D_m) (/ h l))))
   (if (<= d -5.1e+120)
     (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
     (if (<= d -2.15e-305)
       (*
        (* t_0 (/ (sqrt (- d)) (sqrt (- h))))
        (- 1.0 (* (* (* 0.5 (* D_m -0.5)) (/ M_m d)) t_1)))
       (if (<= d 6.9e-193)
         (*
          (* (sqrt (/ d (* h l))) (sqrt d))
          (-
           1.0
           (*
            (*
             (/ 1.0 2.0)
             (* (* (* (/ D_m (+ d d)) M_m) (* D_m M_m)) (/ 0.5 d)))
            (/ h l))))
         (*
          (* t_0 (sqrt (/ d h)))
          (- 1.0 (* (* (/ M_m d) (* -0.25 D_m)) t_1))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = ((M_m * (-0.5 / d)) * D_m) * (h / l);
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (d <= -2.15e-305) {
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * t_1));
	} else if (d <= 6.9e-193) {
		tmp = (sqrt((d / (h * l))) * sqrt(d)) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	} else {
		tmp = (t_0 * sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * t_1));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = ((m_m * ((-0.5d0) / d)) * d_m) * (h / l)
    if (d <= (-5.1d+120)) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else if (d <= (-2.15d-305)) then
        tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0d0 - (((0.5d0 * (d_m * (-0.5d0))) * (m_m / d)) * t_1))
    else if (d <= 6.9d-193) then
        tmp = (sqrt((d / (h * l))) * sqrt(d)) * (1.0d0 - (((1.0d0 / 2.0d0) * ((((d_m / (d + d)) * m_m) * (d_m * m_m)) * (0.5d0 / d))) * (h / l)))
    else
        tmp = (t_0 * sqrt((d / h))) * (1.0d0 - (((m_m / d) * ((-0.25d0) * d_m)) * t_1))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = ((M_m * (-0.5 / d)) * D_m) * (h / l);
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else if (d <= -2.15e-305) {
		tmp = (t_0 * (Math.sqrt(-d) / Math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * t_1));
	} else if (d <= 6.9e-193) {
		tmp = (Math.sqrt((d / (h * l))) * Math.sqrt(d)) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	} else {
		tmp = (t_0 * Math.sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * t_1));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = ((M_m * (-0.5 / d)) * D_m) * (h / l)
	tmp = 0
	if d <= -5.1e+120:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	elif d <= -2.15e-305:
		tmp = (t_0 * (math.sqrt(-d) / math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * t_1))
	elif d <= 6.9e-193:
		tmp = (math.sqrt((d / (h * l))) * math.sqrt(d)) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)))
	else:
		tmp = (t_0 * math.sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * t_1))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(Float64(M_m * Float64(-0.5 / d)) * D_m) * Float64(h / l))
	tmp = 0.0
	if (d <= -5.1e+120)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= -2.15e-305)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(D_m * -0.5)) * Float64(M_m / d)) * t_1)));
	elseif (d <= 6.9e-193)
		tmp = Float64(Float64(sqrt(Float64(d / Float64(h * l))) * sqrt(d)) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(Float64(Float64(Float64(D_m / Float64(d + d)) * M_m) * Float64(D_m * M_m)) * Float64(0.5 / d))) * Float64(h / l))));
	else
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(M_m / d) * Float64(-0.25 * D_m)) * t_1)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = ((M_m * (-0.5 / d)) * D_m) * (h / l);
	tmp = 0.0;
	if (d <= -5.1e+120)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	elseif (d <= -2.15e-305)
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * t_1));
	elseif (d <= 6.9e-193)
		tmp = (sqrt((d / (h * l))) * sqrt(d)) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	else
		tmp = (t_0 * sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M$95$m * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.1e+120], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.15e-305], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.5 * N[(D$95$m * -0.5), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.9e-193], N[(N[(N[Sqrt[N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.10000000000000027e120

   1. Initial program 66.7%

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

   3. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

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

   8. Applied rewrites 25.5%

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

   if -5.10000000000000027e120 < d < -2.1500000000000001e-305

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-neg.f64 38.6

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

   9. Applied rewrites 38.6%

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

   if -2.1500000000000001e-305 < d < 6.8999999999999998e-193

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/r* N/A

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

      22. lift-/.f64 N/A

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

      23. lower-/.f64 65.0

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

      24. lift-/.f64 N/A

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

      25. metadata-eval 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-prod N/A

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

      12. lower-unsound-sqrt.f64 N/A

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

      13. lower-unsound-*.f64 N/A

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

      14. lower-unsound-sqrt.f64 31.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 31.9

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

   7. Applied rewrites 31.9%

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

   if 6.8999999999999998e-193 < d

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.4

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 68.4

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

   9. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{M}{d} \cdot \left(-0.25 \cdot D\right)\right)} \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -5.1 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-305}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D\_m \cdot -0.5\right)\right) \cdot \frac{M\_m}{d}\right) \cdot \left(\left(\left(M\_m \cdot \frac{-0.5}{d}\right) \cdot D\_m\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\left(\frac{D\_m}{d + d} \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{0.5}{d}\right)\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= d -5.1e+120)
     (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
     (if (<= d -2.15e-305)
       (*
        (* t_0 (/ (sqrt (- d)) (sqrt (- h))))
        (-
         1.0
         (*
          (* (* 0.5 (* D_m -0.5)) (/ M_m d))
          (* (* (* M_m (/ -0.5 d)) D_m) (/ h l)))))
       (*
        (* t_0 (/ (sqrt d) (sqrt h)))
        (-
         1.0
         (*
          (* (/ 1.0 2.0) (* (* (* (/ D_m (+ d d)) M_m) (* D_m M_m)) (/ 0.5 d)))
          (/ h l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (d <= -2.15e-305) {
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (d <= (-5.1d+120)) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else if (d <= (-2.15d-305)) then
        tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0d0 - (((0.5d0 * (d_m * (-0.5d0))) * (m_m / d)) * (((m_m * ((-0.5d0) / d)) * d_m) * (h / l))))
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((((d_m / (d + d)) * m_m) * (d_m * m_m)) * (0.5d0 / d))) * (h / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else if (d <= -2.15e-305) {
		tmp = (t_0 * (Math.sqrt(-d) / Math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	} else {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if d <= -5.1e+120:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	elif d <= -2.15e-305:
		tmp = (t_0 * (math.sqrt(-d) / math.sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))))
	else:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -5.1e+120)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= -2.15e-305)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(D_m * -0.5)) * Float64(M_m / d)) * Float64(Float64(Float64(M_m * Float64(-0.5 / d)) * D_m) * Float64(h / l)))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(Float64(Float64(Float64(D_m / Float64(d + d)) * M_m) * Float64(D_m * M_m)) * Float64(0.5 / d))) * Float64(h / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -5.1e+120)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	elseif (d <= -2.15e-305)
		tmp = (t_0 * (sqrt(-d) / sqrt(-h))) * (1.0 - (((0.5 * (D_m * -0.5)) * (M_m / d)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	else
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) * (0.5 / d))) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.1e+120], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.15e-305], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.5 * N[(D$95$m * -0.5), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.10000000000000027e120

   1. Initial program 66.7%

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

   3. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

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

   8. Applied rewrites 25.5%

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

   if -5.10000000000000027e120 < d < -2.1500000000000001e-305

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-neg.f64 38.6

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

   9. Applied rewrites 38.6%

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

   if -2.1500000000000001e-305 < d

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/r* N/A

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

      22. lift-/.f64 N/A

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

      23. lower-/.f64 65.0

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

      24. lift-/.f64 N/A

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

      25. metadata-eval 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-unsound-sqrt.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-sqrt.f64 37.3

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

   7. Applied rewrites 37.3%

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

Alternative 5: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.1 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{M\_m}{d} \cdot \left(-0.25 \cdot D\_m\right)\right) \cdot \left(\left(\left(M\_m \cdot \frac{-0.5}{d}\right) \cdot D\_m\right) \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5.1e+120)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (-
     1.0
     (* (* (/ M_m d) (* -0.25 D_m)) (* (* (* M_m (/ -0.5 d)) D_m) (/ h l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5.1d+120)) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((m_m / d) * ((-0.25d0) * d_m)) * (((m_m * ((-0.5d0) / d)) * d_m) * (h / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.1e+120) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5.1e+120:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5.1e+120)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(M_m / d) * Float64(-0.25 * D_m)) * Float64(Float64(Float64(M_m * Float64(-0.5 / d)) * D_m) * Float64(h / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5.1e+120)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((M_m / d) * (-0.25 * D_m)) * (((M_m * (-0.5 / d)) * D_m) * (h / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5.1e+120], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.10000000000000027e120

   1. Initial program 66.7%

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

   3. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

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

   8. Applied rewrites 25.5%

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

   if -5.10000000000000027e120 < d

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.4

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 68.4

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

   9. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{M}{d} \cdot \left(-0.25 \cdot D\right)\right)} \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4.35 \cdot 10^{+167}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{M\_m \cdot \left(\left(\left(\frac{-0.5}{d} \cdot M\_m\right) \cdot D\_m\right) \cdot h\right)}{d \cdot \ell} \cdot \left(-0.25 \cdot D\_m\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -4.35e+167)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (*
    (*
     (-
      1.0
      (* (/ (* M_m (* (* (* (/ -0.5 d) M_m) D_m) h)) (* d l)) (* -0.25 D_m)))
     (sqrt (/ d h)))
    (sqrt (/ d l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -4.35e+167) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = ((1.0 - (((M_m * ((((-0.5 / d) * M_m) * D_m) * h)) / (d * l)) * (-0.25 * D_m))) * sqrt((d / h))) * sqrt((d / l));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-4.35d+167)) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = ((1.0d0 - (((m_m * (((((-0.5d0) / d) * m_m) * d_m) * h)) / (d * l)) * ((-0.25d0) * d_m))) * sqrt((d / h))) * sqrt((d / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -4.35e+167) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = ((1.0 - (((M_m * ((((-0.5 / d) * M_m) * D_m) * h)) / (d * l)) * (-0.25 * D_m))) * Math.sqrt((d / h))) * Math.sqrt((d / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -4.35e+167:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = ((1.0 - (((M_m * ((((-0.5 / d) * M_m) * D_m) * h)) / (d * l)) * (-0.25 * D_m))) * math.sqrt((d / h))) * math.sqrt((d / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -4.35e+167)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(M_m * Float64(Float64(Float64(Float64(-0.5 / d) * M_m) * D_m) * h)) / Float64(d * l)) * Float64(-0.25 * D_m))) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -4.35e+167)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = ((1.0 - (((M_m * ((((-0.5 / d) * M_m) * D_m) * h)) / (d * l)) * (-0.25 * D_m))) * sqrt((d / h))) * sqrt((d / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -4.35e+167], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(M$95$m * N[(N[(N[(N[(-0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.3499999999999999e167

   1. Initial program 66.7%

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

   3. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

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

   8. Applied rewrites 25.5%

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

   if -4.3499999999999999e167 < d

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 68.4%

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

   9. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{M \cdot \left(\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right) \cdot h\right)}{d \cdot \ell} \cdot \left(-0.25 \cdot D\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+117}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\left(\frac{\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot D\_m}{d} \cdot 0.25\right) \cdot \frac{-0.5}{d}, \frac{h}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.02e+120)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (if (<= d 1.6e+117)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (fma
       (* (* (/ (* (* (* M_m M_m) D_m) D_m) d) 0.25) (/ -0.5 d))
       (/ h l)
       1.0))
     (/ (* d (sqrt (/ 1.0 h))) (sqrt l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.02e+120) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else if (d <= 1.6e+117) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * fma(((((((M_m * M_m) * D_m) * D_m) / d) * 0.25) * (-0.5 / d)), (h / l), 1.0);
	} else {
		tmp = (d * sqrt((1.0 / h))) / sqrt(l);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.02e+120)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (d <= 1.6e+117)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * D_m) / d) * 0.25) * Float64(-0.5 / d)), Float64(h / l), 1.0));
	else
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / h))) / sqrt(l));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.02e+120], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.6e+117], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.01999999999999997e120

   1. Initial program 66.7%

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

   3. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

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

   8. Applied rewrites 25.5%

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

   if -1.01999999999999997e120 < d < 1.60000000000000002e117

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 61.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 59.9%

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

   if 1.60000000000000002e117 < d

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg N/A

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

      9. sqr-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

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

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

      25. lower-*.f64 N/A

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

      26. metadata-eval 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 24.7

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

   10. Applied rewrites 24.7%

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

Alternative 8: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-146}:\\ \;\;\;\;t\_1 \cdot \left(1 - \frac{\left(\left(\left(\left(\frac{D\_m}{d + d} \cdot M\_m\right) \cdot M\_m\right) \cdot D\_m\right) \cdot 0.25\right) \cdot h}{d \cdot \ell}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+283}:\\ \;\;\;\;t\_1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= t_0 -1e-146)
     (*
      t_1
      (-
       1.0
       (/ (* (* (* (* (* (/ D_m (+ d d)) M_m) M_m) D_m) 0.25) h) (* d l))))
     (if (<= t_0 1e+283) (* t_1 1.0) (* (sqrt (* (/ d (* h l)) d)) (/ l l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = t_1 * (1.0 - (((((((D_m / (d + d)) * M_m) * M_m) * D_m) * 0.25) * h) / (d * l)));
	} else if (t_0 <= 1e+283) {
		tmp = t_1 * 1.0;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / l)) * sqrt((d / h))
    if (t_0 <= (-1d-146)) then
        tmp = t_1 * (1.0d0 - (((((((d_m / (d + d)) * m_m) * m_m) * d_m) * 0.25d0) * h) / (d * l)))
    else if (t_0 <= 1d+283) then
        tmp = t_1 * 1.0d0
    else
        tmp = sqrt(((d / (h * l)) * d)) * (l / l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = t_1 * (1.0 - (((((((D_m / (d + d)) * M_m) * M_m) * D_m) * 0.25) * h) / (d * l)));
	} else if (t_0 <= 1e+283) {
		tmp = t_1 * 1.0;
	} else {
		tmp = Math.sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / l)) * math.sqrt((d / h))
	tmp = 0
	if t_0 <= -1e-146:
		tmp = t_1 * (1.0 - (((((((D_m / (d + d)) * M_m) * M_m) * D_m) * 0.25) * h) / (d * l)))
	elif t_0 <= 1e+283:
		tmp = t_1 * 1.0
	else:
		tmp = math.sqrt(((d / (h * l)) * d)) * (l / l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (t_0 <= -1e-146)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(D_m / Float64(d + d)) * M_m) * M_m) * D_m) * 0.25) * h) / Float64(d * l))));
	elseif (t_0 <= 1e+283)
		tmp = Float64(t_1 * 1.0);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(l / l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / l)) * sqrt((d / h));
	tmp = 0.0;
	if (t_0 <= -1e-146)
		tmp = t_1 * (1.0 - (((((((D_m / (d + d)) * M_m) * M_m) * D_m) * 0.25) * h) / (d * l)));
	elseif (t_0 <= 1e+283)
		tmp = t_1 * 1.0;
	else
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-146], N[(t$95$1 * N[(1.0 - N[(N[(N[(N[(N[(N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(t$95$1 * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(l / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000003e-146

   1. Initial program 66.7%

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

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

      3. lower-*.f64 66.7

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.7

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 61.3%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot 0.5\right) \cdot \frac{0.5}{d}\right)} \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{2}}{d}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{2}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}{d}} \cdot \frac{h}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot h}{d \cdot \ell}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot h}{d \cdot \ell}}\right) \]

   9. Applied rewrites 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\left(\frac{D}{d + d} \cdot M\right) \cdot M\right) \cdot D\right) \cdot 0.25\right) \cdot h}{d \cdot \ell}\right)} \]

   if -1.00000000000000003e-146 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999955e282

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \frac{D \cdot M}{-2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)}\right) \]

   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 38.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.99999999999999955e282 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-146}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.5, \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot D\_m\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+283}:\\ \;\;\;\;t\_1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= t_0 -1e-146)
     (*
      t_1
      (fma
       -0.5
       (/ (* (* (* (* M_m M_m) D_m) D_m) h) (* (* 4.0 (* d d)) l))
       1.0))
     (if (<= t_0 1e+283) (* t_1 1.0) (* (sqrt (* (/ d (* h l)) d)) (/ l l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = t_1 * fma(-0.5, (((((M_m * M_m) * D_m) * D_m) * h) / ((4.0 * (d * d)) * l)), 1.0);
	} else if (t_0 <= 1e+283) {
		tmp = t_1 * 1.0;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (t_0 <= -1e-146)
		tmp = Float64(t_1 * fma(-0.5, Float64(Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * D_m) * h) / Float64(Float64(4.0 * Float64(d * d)) * l)), 1.0));
	elseif (t_0 <= 1e+283)
		tmp = Float64(t_1 * 1.0);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(l / l));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-146], N[(t$95$1 * N[(-0.5 * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(t$95$1 * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(l / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000003e-146

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 50.8%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot D\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell}, 1\right)} \]

   if -1.00000000000000003e-146 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999955e282

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \frac{D \cdot M}{-2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)}\right) \]

   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 38.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.99999999999999955e282 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(\left(1 - \left(\left(\frac{\left(D\_m \cdot D\_m\right) \cdot M\_m}{d \cdot d} \cdot 0.125\right) \cdot M\_m\right) \cdot \frac{h}{\ell}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+283}:\\ \;\;\;\;\left(t\_1 \cdot t\_2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h))))
   (if (<= t_0 -1e-146)
     (*
      (*
       (- 1.0 (* (* (* (/ (* (* D_m D_m) M_m) (* d d)) 0.125) M_m) (/ h l)))
       t_1)
      t_2)
     (if (<= t_0 1e+283)
       (* (* t_1 t_2) 1.0)
       (* (sqrt (* (/ d (* h l)) d)) (/ l l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2;
	} else if (t_0 <= 1e+283) {
		tmp = (t_1 * t_2) * 1.0;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / l))
    t_2 = sqrt((d / h))
    if (t_0 <= (-1d-146)) then
        tmp = ((1.0d0 - ((((((d_m * d_m) * m_m) / (d * d)) * 0.125d0) * m_m) * (h / l))) * t_1) * t_2
    else if (t_0 <= 1d+283) then
        tmp = (t_1 * t_2) * 1.0d0
    else
        tmp = sqrt(((d / (h * l)) * d)) * (l / l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2;
	} else if (t_0 <= 1e+283) {
		tmp = (t_1 * t_2) * 1.0;
	} else {
		tmp = Math.sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if t_0 <= -1e-146:
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2
	elif t_0 <= 1e+283:
		tmp = (t_1 * t_2) * 1.0
	else:
		tmp = math.sqrt(((d / (h * l)) * d)) * (l / l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_0 <= -1e-146)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * M_m) / Float64(d * d)) * 0.125) * M_m) * Float64(h / l))) * t_1) * t_2);
	elseif (t_0 <= 1e+283)
		tmp = Float64(Float64(t_1 * t_2) * 1.0);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(l / l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / l));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (t_0 <= -1e-146)
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2;
	elseif (t_0 <= 1e+283)
		tmp = (t_1 * t_2) * 1.0;
	else
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-146], N[(N[(N[(1.0 - N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(N[(t$95$1 * t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(l / l), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000003e-146

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\left(\left(1 - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if -1.00000000000000003e-146 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999955e282

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \frac{D \cdot M}{-2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)}\right) \]

   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 38.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.99999999999999955e282 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(\left(1 - \left(\left(\frac{\left(D\_m \cdot D\_m\right) \cdot M\_m}{d \cdot d} \cdot 0.125\right) \cdot M\_m\right) \cdot \frac{h}{\ell}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+283}:\\ \;\;\;\;\left(t\_2 \cdot t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -1e-146)
     (*
      (*
       (- 1.0 (* (* (* (/ (* (* D_m D_m) M_m) (* d d)) 0.125) M_m) (/ h l)))
       t_1)
      t_2)
     (if (<= t_0 1e+283)
       (* (* t_2 t_1) 1.0)
       (* (sqrt (* (/ d (* h l)) d)) (/ l l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2;
	} else if (t_0 <= 1e+283) {
		tmp = (t_2 * t_1) * 1.0;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-1d-146)) then
        tmp = ((1.0d0 - ((((((d_m * d_m) * m_m) / (d * d)) * 0.125d0) * m_m) * (h / l))) * t_1) * t_2
    else if (t_0 <= 1d+283) then
        tmp = (t_2 * t_1) * 1.0d0
    else
        tmp = sqrt(((d / (h * l)) * d)) * (l / l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2;
	} else if (t_0 <= 1e+283) {
		tmp = (t_2 * t_1) * 1.0;
	} else {
		tmp = Math.sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -1e-146:
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2
	elif t_0 <= 1e+283:
		tmp = (t_2 * t_1) * 1.0
	else:
		tmp = math.sqrt(((d / (h * l)) * d)) * (l / l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -1e-146)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * M_m) / Float64(d * d)) * 0.125) * M_m) * Float64(h / l))) * t_1) * t_2);
	elseif (t_0 <= 1e+283)
		tmp = Float64(Float64(t_2 * t_1) * 1.0);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(l / l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -1e-146)
		tmp = ((1.0 - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * (h / l))) * t_1) * t_2;
	elseif (t_0 <= 1e+283)
		tmp = (t_2 * t_1) * 1.0;
	else
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-146], N[(N[(N[(1.0 - N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(N[(t$95$2 * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(l / l), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000003e-146

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\left(\left(1 - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -1.00000000000000003e-146 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999955e282

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \frac{D \cdot M}{-2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)}\right) \]

   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 38.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.99999999999999955e282 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \mathbf{elif}\;t\_0 \leq 10^{+283}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-146)
     (*
      (fma
       (* (* (* (* M_m D_m) (* M_m D_m)) (/ 0.25 (* (* d d) l))) h)
       -0.5
       1.0)
      (sqrt (* (/ d (* l h)) d)))
     (if (<= t_0 1e+283)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (* (sqrt (* (/ d (* h l)) d)) (/ l l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = fma(((((M_m * D_m) * (M_m * D_m)) * (0.25 / ((d * d) * l))) * h), -0.5, 1.0) * sqrt(((d / (l * h)) * d));
	} else if (t_0 <= 1e+283) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-146)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(0.25 / Float64(Float64(d * d) * l))) * h), -0.5, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	elseif (t_0 <= 1e+283)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(l / l));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-146], N[(N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.25 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(l / l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000003e-146

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\left(D \cdot D\right) \cdot M\right)} \cdot M\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)} \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      5. unswap-sqr N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(D \cdot M\right) \cdot \color{blue}{\left(D \cdot M\right)}\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      8. lower-*.f64 44.0

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      11. lift-*.f64 44.0

          \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \left(D \cdot M\right)\right) \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(M \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(M \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      14. lift-*.f64 44.0

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(M \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}\right) \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

   7. Applied rewrites 44.0%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

   if -1.00000000000000003e-146 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999955e282

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \frac{D \cdot M}{-2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)}\right) \]

   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 38.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.99999999999999955e282 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(D\_m \cdot \left(\frac{0.25}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m\right)\right)\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \mathbf{elif}\;t\_0 \leq 10^{+283}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-146)
     (*
      (fma
       (* (* (* M_m D_m) (* D_m (* (/ 0.25 (* (* d d) l)) M_m))) h)
       -0.5
       1.0)
      (sqrt (* (/ d (* l h)) d)))
     (if (<= t_0 1e+283)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (* (sqrt (* (/ d (* h l)) d)) (/ l l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-146) {
		tmp = fma((((M_m * D_m) * (D_m * ((0.25 / ((d * d) * l)) * M_m))) * h), -0.5, 1.0) * sqrt(((d / (l * h)) * d));
	} else if (t_0 <= 1e+283) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (l / l);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-146)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * D_m) * Float64(D_m * Float64(Float64(0.25 / Float64(Float64(d * d) * l)) * M_m))) * h), -0.5, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	elseif (t_0 <= 1e+283)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(l / l));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-146], N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(D$95$m * N[(N[(0.25 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(l / l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000003e-146

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)} \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)} \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right)} \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(D \cdot D\right) \cdot M\right)} \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(D \cdot \left(D \cdot M\right)\right)} \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(D \cdot \color{blue}{\left(D \cdot M\right)}\right) \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(D \cdot M\right) \cdot D\right)} \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right)} \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right)} \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \left(D \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \left(D \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(M \cdot D\right)} \cdot \left(D \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell}\right)\right)}\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(D \cdot \color{blue}{\left(\frac{\frac{1}{4}}{\left(d \cdot d\right) \cdot \ell} \cdot M\right)}\right)\right) \cdot h, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      16. lower-*.f64 45.6

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot \left(D \cdot \color{blue}{\left(\frac{0.25}{\left(d \cdot d\right) \cdot \ell} \cdot M\right)}\right)\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

   7. Applied rewrites 45.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(D \cdot \left(\frac{0.25}{\left(d \cdot d\right) \cdot \ell} \cdot M\right)\right)\right)} \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

   if -1.00000000000000003e-146 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999955e282

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \frac{D \cdot M}{-2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)}\right) \]

   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 38.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.99999999999999955e282 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h \cdot \ell} \cdot d}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-67}:\\ \;\;\;\;t\_1 \cdot \frac{\ell - \left(\left(\left(D\_m \cdot \frac{M\_m \cdot D\_m}{d \cdot d}\right) \cdot 0.125\right) \cdot M\_m\right) \cdot h}{\ell}\\ \mathbf{elif}\;t\_0 \leq 10^{+283}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\ell}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (* (/ d (* h l)) d))))
   (if (<= t_0 -2e-67)
     (*
      t_1
      (/ (- l (* (* (* (* D_m (/ (* M_m D_m) (* d d))) 0.125) M_m) h)) l))
     (if (<= t_0 1e+283)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (* t_1 (/ l l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt(((d / (h * l)) * d));
	double tmp;
	if (t_0 <= -2e-67) {
		tmp = t_1 * ((l - ((((D_m * ((M_m * D_m) / (d * d))) * 0.125) * M_m) * h)) / l);
	} else if (t_0 <= 1e+283) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1 * (l / l);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt(((d / (h * l)) * d))
    if (t_0 <= (-2d-67)) then
        tmp = t_1 * ((l - ((((d_m * ((m_m * d_m) / (d * d))) * 0.125d0) * m_m) * h)) / l)
    else if (t_0 <= 1d+283) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1 * (l / l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt(((d / (h * l)) * d));
	double tmp;
	if (t_0 <= -2e-67) {
		tmp = t_1 * ((l - ((((D_m * ((M_m * D_m) / (d * d))) * 0.125) * M_m) * h)) / l);
	} else if (t_0 <= 1e+283) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1 * (l / l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (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_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt(((d / (h * l)) * d))
	tmp = 0
	if t_0 <= -2e-67:
		tmp = t_1 * ((l - ((((D_m * ((M_m * D_m) / (d * d))) * 0.125) * M_m) * h)) / l)
	elif t_0 <= 1e+283:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1 * (l / l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(Float64(d / Float64(h * l)) * d))
	tmp = 0.0
	if (t_0 <= -2e-67)
		tmp = Float64(t_1 * Float64(Float64(l - Float64(Float64(Float64(Float64(D_m * Float64(Float64(M_m * D_m) / Float64(d * d))) * 0.125) * M_m) * h)) / l));
	elseif (t_0 <= 1e+283)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(t_1 * Float64(l / l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt(((d / (h * l)) * d));
	tmp = 0.0;
	if (t_0 <= -2e-67)
		tmp = t_1 * ((l - ((((D_m * ((M_m * D_m) / (d * d))) * 0.125) * M_m) * h)) / l);
	elseif (t_0 <= 1e+283)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1 * (l / l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -2e-67], N[(t$95$1 * N[(N[(l - N[(N[(N[(N[(D$95$m * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$1 * N[(l / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999989e-67

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\color{blue}{\frac{\left(D \cdot D\right) \cdot M}{d \cdot d}} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\color{blue}{\left(D \cdot D\right) \cdot M}}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot M}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\color{blue}{D \cdot \left(D \cdot M\right)}}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{D \cdot \color{blue}{\left(D \cdot M\right)}}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\color{blue}{\left(D \cdot \frac{D \cdot M}{d \cdot d}\right)} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\color{blue}{\left(D \cdot \frac{D \cdot M}{d \cdot d}\right)} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      8. lower-/.f64 45.4

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\left(D \cdot \color{blue}{\frac{D \cdot M}{d \cdot d}}\right) \cdot 0.125\right) \cdot M\right) \cdot h}{\ell} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\left(D \cdot \frac{\color{blue}{D \cdot M}}{d \cdot d}\right) \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\left(D \cdot \frac{\color{blue}{M \cdot D}}{d \cdot d}\right) \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      11. lower-*.f64 45.4

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\left(D \cdot \frac{\color{blue}{M \cdot D}}{d \cdot d}\right) \cdot 0.125\right) \cdot M\right) \cdot h}{\ell} \]

   7. Applied rewrites 45.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\color{blue}{\left(D \cdot \frac{M \cdot D}{d \cdot d}\right)} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell} \]

   if -1.99999999999999989e-67 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999955e282

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \frac{D \cdot M}{-2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{-2 \cdot d}\right) \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot -0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(M \cdot \frac{-0.5}{d}\right) \cdot D\right) \cdot \frac{h}{\ell}\right)}\right) \]

   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 38.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 9.99999999999999955e282 < (*.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 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\color{blue}{\ell}}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-305}:\\ \;\;\;\;-1 \cdot t\_0\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;t\_0 \cdot \frac{\ell - \left(\left(\frac{\left(D\_m \cdot D\_m\right) \cdot M\_m}{d \cdot d} \cdot 0.125\right) \cdot M\_m\right) \cdot h}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ 1.0 (* h l))))))
   (if (<= l 1.4e-305)
     (* -1.0 t_0)
     (if (<= l 6.2e+82)
       (*
        t_0
        (/ (- l (* (* (* (/ (* (* D_m D_m) M_m) (* d d)) 0.125) M_m) h)) l))
       (/ (* d (sqrt (/ 1.0 h))) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 1.4e-305) {
		tmp = -1.0 * t_0;
	} else if (l <= 6.2e+82) {
		tmp = t_0 * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l);
	} else {
		tmp = (d * sqrt((1.0 / h))) / sqrt(l);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * sqrt((1.0d0 / (h * l)))
    if (l <= 1.4d-305) then
        tmp = (-1.0d0) * t_0
    else if (l <= 6.2d+82) then
        tmp = t_0 * ((l - ((((((d_m * d_m) * m_m) / (d * d)) * 0.125d0) * m_m) * h)) / l)
    else
        tmp = (d * sqrt((1.0d0 / h))) / sqrt(l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= 1.4e-305) {
		tmp = -1.0 * t_0;
	} else if (l <= 6.2e+82) {
		tmp = t_0 * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l);
	} else {
		tmp = (d * Math.sqrt((1.0 / h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d * math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= 1.4e-305:
		tmp = -1.0 * t_0
	elif l <= 6.2e+82:
		tmp = t_0 * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l)
	else:
		tmp = (d * math.sqrt((1.0 / h))) / math.sqrt(l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
	tmp = 0.0
	if (l <= 1.4e-305)
		tmp = Float64(-1.0 * t_0);
	elseif (l <= 6.2e+82)
		tmp = Float64(t_0 * Float64(Float64(l - Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * M_m) / Float64(d * d)) * 0.125) * M_m) * h)) / l));
	else
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / h))) / sqrt(l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d * sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= 1.4e-305)
		tmp = -1.0 * t_0;
	elseif (l <= 6.2e+82)
		tmp = t_0 * ((l - ((((((D_m * D_m) * M_m) / (d * d)) * 0.125) * M_m) * h)) / l);
	else
		tmp = (d * sqrt((1.0 / h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$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, 1.4e-305], N[(-1.0 * t$95$0), $MachinePrecision], If[LessEqual[l, 6.2e+82], N[(t$95$0 * N[(N[(l - N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.40000000000000007e-305

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. Applied rewrites 25.5%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 1.40000000000000007e-305 < l < 6.20000000000000065e82

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot \frac{1}{8}\right) \cdot M\right) \cdot h}{\ell} \]

      4. lower-*.f64 27.8

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell} \]

   8. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell} \]

   if 6.20000000000000065e82 < l

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \sqrt{d \cdot \frac{d}{h}}}{\sqrt{\ell}}} \]

   8. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\color{blue}{\ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

      5. lower-sqrt.f64 24.7

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

   10. Applied rewrites 24.7%

       \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 2.3 \cdot 10^{-297}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h 2.3e-297)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (/ (* d (sqrt (/ 1.0 h))) (sqrt l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 2.3e-297) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * sqrt((1.0 / h))) / sqrt(l);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= 2.3d-297) 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

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 2.3e-297) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * Math.sqrt((1.0 / h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= 2.3e-297:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d * math.sqrt((1.0 / h))) / math.sqrt(l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 2.3e-297)
		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

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= 2.3e-297)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d * sqrt((1.0 / h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 2.3e-297], 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 h < 2.2999999999999999e-297

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}}{\ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h\right)}}{\ell} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \frac{\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \frac{\ell - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot h}{\ell}} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. 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 25.5

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. Applied rewrites 25.5%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 2.2999999999999999e-297 < h

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 66.7

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. sqr-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      24. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      25. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

      26. metadata-eval 66.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \sqrt{d \cdot \frac{d}{h}}}{\sqrt{\ell}}} \]

   8. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\color{blue}{\ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

      5. lower-sqrt.f64 24.7

         \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

   10. Applied rewrites 24.7%

       \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 24.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (/ (* d (sqrt (/ 1.0 h))) (sqrt l)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return (d * sqrt((1.0 / h))) / sqrt(l);
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_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_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_m
    real(8), intent (in) :: d_m
    code = (d * sqrt((1.0d0 / h))) / sqrt(l)
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return (d * Math.sqrt((1.0 / h))) / Math.sqrt(l);
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return (d * math.sqrt((1.0 / h))) / math.sqrt(l)

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(Float64(d * sqrt(Float64(1.0 / h))) / sqrt(l))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = (d * sqrt((1.0 / h))) / sqrt(l);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(N[(d * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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

      \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. lower-*.f64 66.7

      \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. metadata-eval N/A

      \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. unpow1/2 N/A

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. lower-sqrt.f64 66.7

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. lift-pow.f64 N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. unpow1/2 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lower-sqrt.f64 66.7

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 66.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
4. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   4. frac-2neg N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. distribute-frac-neg N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

   7. frac-2neg N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(M \cdot D\right)}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

   8. distribute-frac-neg N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   9. sqr-neg N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   13. *-commutative N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   16. distribute-lft-neg-in N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   18. metadata-eval N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{\color{blue}{-2} \cdot d} \cdot \frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right) \cdot \frac{h}{\ell}\right) \]

   20. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   21. *-commutative N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   22. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   23. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

   24. distribute-lft-neg-in N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

   25. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \]

   26. metadata-eval 66.7

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{\color{blue}{-2} \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

5. Applied rewrites 66.7%

   \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

7. Applied rewrites 19.9%

   \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{\left(D \cdot D\right) \cdot M}{d \cdot d} \cdot 0.125\right) \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \sqrt{d \cdot \frac{d}{h}}}{\sqrt{\ell}}} \]

8. Taylor expanded in d around inf

   \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\color{blue}{\sqrt{\ell}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\color{blue}{\ell}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

   5. lower-sqrt.f64 24.7

      \[\leadsto \frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]

10. Applied rewrites 24.7%

    \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025164 
(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)))))