Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 74.8%

Time: 11.3s

Alternatives: 20

Speedup: 1.6×

Specification

Math

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

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: 66.9% accurate, 1.0× speedup

Math

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

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: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{D}{d + d}\\ t_2 := t\_1 \cdot M\\ t_3 := 1 - t\_2 \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\\ t_4 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{t\_4}{\sqrt{-\ell}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{elif}\;h \leq -2.1 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(t\_4 \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell} \cdot d}}{\sqrt{h}} \cdot \left(1 - t\_2 \cdot \frac{\left(h \cdot \left(M \cdot t\_1\right)\right) \cdot 0.5}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (/ D (+ d d)))
        (t_2 (* t_1 M))
        (t_3 (- 1.0 (* t_2 (* (/ (* (* D M) h) (* (+ d d) l)) 0.5))))
        (t_4 (sqrt (- d))))
   (if (<= h -2e-44)
     (* (* (/ t_4 (sqrt (- l))) t_0) t_3)
     (if (<= h -2.1e-307)
       (*
        (* (* t_4 (sqrt (/ -1.0 h))) (pow (/ d l) (/ 1.0 2.0)))
        (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
       (if (<= h 2e+53)
         (* (* (/ (sqrt d) (sqrt l)) t_0) t_3)
         (*
          (/ (sqrt (* (/ d l) d)) (sqrt h))
          (- 1.0 (* t_2 (/ (* (* h (* M t_1)) 0.5) l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = D / (d + d);
	double t_2 = t_1 * M;
	double t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	double t_4 = sqrt(-d);
	double tmp;
	if (h <= -2e-44) {
		tmp = ((t_4 / sqrt(-l)) * t_0) * t_3;
	} else if (h <= -2.1e-307) {
		tmp = ((t_4 * sqrt((-1.0 / h))) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (h <= 2e+53) {
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3;
	} else {
		tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = d_1 / (d + d)
    t_2 = t_1 * m
    t_3 = 1.0d0 - (t_2 * ((((d_1 * m) * h) / ((d + d) * l)) * 0.5d0))
    t_4 = sqrt(-d)
    if (h <= (-2d-44)) then
        tmp = ((t_4 / sqrt(-l)) * t_0) * t_3
    else if (h <= (-2.1d-307)) then
        tmp = ((t_4 * sqrt(((-1.0d0) / h))) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    else if (h <= 2d+53) then
        tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3
    else
        tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0d0 - (t_2 * (((h * (m * t_1)) * 0.5d0) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = D / (d + d);
	double t_2 = t_1 * M;
	double t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	double t_4 = Math.sqrt(-d);
	double tmp;
	if (h <= -2e-44) {
		tmp = ((t_4 / Math.sqrt(-l)) * t_0) * t_3;
	} else if (h <= -2.1e-307) {
		tmp = ((t_4 * Math.sqrt((-1.0 / h))) * 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)));
	} else if (h <= 2e+53) {
		tmp = ((Math.sqrt(d) / Math.sqrt(l)) * t_0) * t_3;
	} else {
		tmp = (Math.sqrt(((d / l) * d)) / Math.sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = D / (d + d)
	t_2 = t_1 * M
	t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5))
	t_4 = math.sqrt(-d)
	tmp = 0
	if h <= -2e-44:
		tmp = ((t_4 / math.sqrt(-l)) * t_0) * t_3
	elif h <= -2.1e-307:
		tmp = ((t_4 * math.sqrt((-1.0 / h))) * 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)))
	elif h <= 2e+53:
		tmp = ((math.sqrt(d) / math.sqrt(l)) * t_0) * t_3
	else:
		tmp = (math.sqrt(((d / l) * d)) / math.sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(D / Float64(d + d))
	t_2 = Float64(t_1 * M)
	t_3 = Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(Float64(D * M) * h) / Float64(Float64(d + d) * l)) * 0.5)))
	t_4 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2e-44)
		tmp = Float64(Float64(Float64(t_4 / sqrt(Float64(-l))) * t_0) * t_3);
	elseif (h <= -2.1e-307)
		tmp = Float64(Float64(Float64(t_4 * sqrt(Float64(-1.0 / h))) * (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))));
	elseif (h <= 2e+53)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(l)) * t_0) * t_3);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(d / l) * d)) / sqrt(h)) * Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(h * Float64(M * t_1)) * 0.5) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = D / (d + d);
	t_2 = t_1 * M;
	t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	t_4 = sqrt(-d);
	tmp = 0.0;
	if (h <= -2e-44)
		tmp = ((t_4 / sqrt(-l)) * t_0) * t_3;
	elseif (h <= -2.1e-307)
		tmp = ((t_4 * sqrt((-1.0 / h))) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	elseif (h <= 2e+53)
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3;
	else
		tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * M), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(t$95$2 * N[(N[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2e-44], N[(N[(N[(t$95$4 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[h, -2.1e-307], N[(N[(N[(t$95$4 * N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $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], If[LessEqual[h, 2e+53], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(N[(N[(h * N[(M * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.99999999999999991e-44

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

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

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

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

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

      7. lower-neg.f64 N/A

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

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

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

      9. lower-neg.f64 36.4%

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

   7. Applied rewrites 36.4%

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

   if -1.99999999999999991e-44 < h < -2.1000000000000001e-307

   1. Initial program 66.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. mult-flip N/A

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

      8. sqrt-prod N/A

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

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

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

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

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

      11. lower-neg.f64 N/A

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

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

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 37.8%

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

   3. Applied rewrites 37.8%

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

   if -2.1000000000000001e-307 < h < 2e53

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 36.4%

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

   7. Applied rewrites 36.4%

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

   if 2e53 < h

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. sqrt-div N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-unsound-sqrt.f64 32.5%

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

   9. Applied rewrites 32.5%

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

Alternative 2: 72.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{D}{d + d}\\ t_2 := t\_1 \cdot M\\ t_3 := 1 - t\_2 \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\\ t_4 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{t\_4}{\sqrt{-\ell}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{elif}\;h \leq -2.1 \cdot 10^{-307}:\\ \;\;\;\;\left(\frac{t\_4}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell} \cdot d}}{\sqrt{h}} \cdot \left(1 - t\_2 \cdot \frac{\left(h \cdot \left(M \cdot t\_1\right)\right) \cdot 0.5}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (/ D (+ d d)))
        (t_2 (* t_1 M))
        (t_3 (- 1.0 (* t_2 (* (/ (* (* D M) h) (* (+ d d) l)) 0.5))))
        (t_4 (sqrt (- d))))
   (if (<= h -2e-44)
     (* (* (/ t_4 (sqrt (- l))) t_0) t_3)
     (if (<= h -2.1e-307)
       (*
        (* (/ t_4 (sqrt (- h))) (pow (/ d l) (/ 1.0 2.0)))
        (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
       (if (<= h 2e+53)
         (* (* (/ (sqrt d) (sqrt l)) t_0) t_3)
         (*
          (/ (sqrt (* (/ d l) d)) (sqrt h))
          (- 1.0 (* t_2 (/ (* (* h (* M t_1)) 0.5) l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = D / (d + d);
	double t_2 = t_1 * M;
	double t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	double t_4 = sqrt(-d);
	double tmp;
	if (h <= -2e-44) {
		tmp = ((t_4 / sqrt(-l)) * t_0) * t_3;
	} else if (h <= -2.1e-307) {
		tmp = ((t_4 / sqrt(-h)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (h <= 2e+53) {
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3;
	} else {
		tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = d_1 / (d + d)
    t_2 = t_1 * m
    t_3 = 1.0d0 - (t_2 * ((((d_1 * m) * h) / ((d + d) * l)) * 0.5d0))
    t_4 = sqrt(-d)
    if (h <= (-2d-44)) then
        tmp = ((t_4 / sqrt(-l)) * t_0) * t_3
    else if (h <= (-2.1d-307)) then
        tmp = ((t_4 / sqrt(-h)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    else if (h <= 2d+53) then
        tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3
    else
        tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0d0 - (t_2 * (((h * (m * t_1)) * 0.5d0) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = D / (d + d);
	double t_2 = t_1 * M;
	double t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	double t_4 = Math.sqrt(-d);
	double tmp;
	if (h <= -2e-44) {
		tmp = ((t_4 / Math.sqrt(-l)) * t_0) * t_3;
	} else if (h <= -2.1e-307) {
		tmp = ((t_4 / Math.sqrt(-h)) * 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)));
	} else if (h <= 2e+53) {
		tmp = ((Math.sqrt(d) / Math.sqrt(l)) * t_0) * t_3;
	} else {
		tmp = (Math.sqrt(((d / l) * d)) / Math.sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = D / (d + d)
	t_2 = t_1 * M
	t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5))
	t_4 = math.sqrt(-d)
	tmp = 0
	if h <= -2e-44:
		tmp = ((t_4 / math.sqrt(-l)) * t_0) * t_3
	elif h <= -2.1e-307:
		tmp = ((t_4 / math.sqrt(-h)) * 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)))
	elif h <= 2e+53:
		tmp = ((math.sqrt(d) / math.sqrt(l)) * t_0) * t_3
	else:
		tmp = (math.sqrt(((d / l) * d)) / math.sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(D / Float64(d + d))
	t_2 = Float64(t_1 * M)
	t_3 = Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(Float64(D * M) * h) / Float64(Float64(d + d) * l)) * 0.5)))
	t_4 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2e-44)
		tmp = Float64(Float64(Float64(t_4 / sqrt(Float64(-l))) * t_0) * t_3);
	elseif (h <= -2.1e-307)
		tmp = Float64(Float64(Float64(t_4 / sqrt(Float64(-h))) * (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))));
	elseif (h <= 2e+53)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(l)) * t_0) * t_3);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(d / l) * d)) / sqrt(h)) * Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(h * Float64(M * t_1)) * 0.5) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = D / (d + d);
	t_2 = t_1 * M;
	t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	t_4 = sqrt(-d);
	tmp = 0.0;
	if (h <= -2e-44)
		tmp = ((t_4 / sqrt(-l)) * t_0) * t_3;
	elseif (h <= -2.1e-307)
		tmp = ((t_4 / sqrt(-h)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	elseif (h <= 2e+53)
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3;
	else
		tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * M), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(t$95$2 * N[(N[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2e-44], N[(N[(N[(t$95$4 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[h, -2.1e-307], N[(N[(N[(t$95$4 / N[Sqrt[(-h)], $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], If[LessEqual[h, 2e+53], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(N[(N[(h * N[(M * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.99999999999999991e-44

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

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

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

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

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

      7. lower-neg.f64 N/A

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

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

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

      9. lower-neg.f64 36.4%

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

   7. Applied rewrites 36.4%

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

   if -1.99999999999999991e-44 < h < -2.1000000000000001e-307

   1. Initial program 66.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

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

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

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

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

      10. lower-neg.f64 N/A

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

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

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

      12. lower-neg.f64 37.8%

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

   3. Applied rewrites 37.8%

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

   if -2.1000000000000001e-307 < h < 2e53

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 36.4%

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

   7. Applied rewrites 36.4%

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

   if 2e53 < h

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. sqrt-div N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-unsound-sqrt.f64 32.5%

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

   9. Applied rewrites 32.5%

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

Alternative 3: 72.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{D}{d + d}\\ t_2 := t\_1 \cdot M\\ t_3 := 1 - t\_2 \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{elif}\;h \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell} \cdot d}}{\sqrt{h}} \cdot \left(1 - t\_2 \cdot \frac{\left(h \cdot \left(M \cdot t\_1\right)\right) \cdot 0.5}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (/ D (+ d d)))
        (t_2 (* t_1 M))
        (t_3 (- 1.0 (* t_2 (* (/ (* (* D M) h) (* (+ d d) l)) 0.5)))))
   (if (<= h -5e-310)
     (* (* (/ (sqrt (- d)) (sqrt (- l))) t_0) t_3)
     (if (<= h 2e+53)
       (* (* (/ (sqrt d) (sqrt l)) t_0) t_3)
       (*
        (/ (sqrt (* (/ d l) d)) (sqrt h))
        (- 1.0 (* t_2 (/ (* (* h (* M t_1)) 0.5) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = D / (d + d);
	double t_2 = t_1 * M;
	double t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	double tmp;
	if (h <= -5e-310) {
		tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * t_3;
	} else if (h <= 2e+53) {
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3;
	} else {
		tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = d_1 / (d + d)
    t_2 = t_1 * m
    t_3 = 1.0d0 - (t_2 * ((((d_1 * m) * h) / ((d + d) * l)) * 0.5d0))
    if (h <= (-5d-310)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * t_3
    else if (h <= 2d+53) then
        tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3
    else
        tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0d0 - (t_2 * (((h * (m * t_1)) * 0.5d0) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = D / (d + d);
	double t_2 = t_1 * M;
	double t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	double tmp;
	if (h <= -5e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-l)) * t_0) * t_3;
	} else if (h <= 2e+53) {
		tmp = ((Math.sqrt(d) / Math.sqrt(l)) * t_0) * t_3;
	} else {
		tmp = (Math.sqrt(((d / l) * d)) / Math.sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = D / (d + d)
	t_2 = t_1 * M
	t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5))
	tmp = 0
	if h <= -5e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * t_0) * t_3
	elif h <= 2e+53:
		tmp = ((math.sqrt(d) / math.sqrt(l)) * t_0) * t_3
	else:
		tmp = (math.sqrt(((d / l) * d)) / math.sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(D / Float64(d + d))
	t_2 = Float64(t_1 * M)
	t_3 = Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(Float64(D * M) * h) / Float64(Float64(d + d) * l)) * 0.5)))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * t_0) * t_3);
	elseif (h <= 2e+53)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(l)) * t_0) * t_3);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(d / l) * d)) / sqrt(h)) * Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(h * Float64(M * t_1)) * 0.5) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = D / (d + d);
	t_2 = t_1 * M;
	t_3 = 1.0 - (t_2 * ((((D * M) * h) / ((d + d) * l)) * 0.5));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * t_3;
	elseif (h <= 2e+53)
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_3;
	else
		tmp = (sqrt(((d / l) * d)) / sqrt(h)) * (1.0 - (t_2 * (((h * (M * t_1)) * 0.5) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * M), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(t$95$2 * N[(N[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[h, 2e+53], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(N[(N[(h * N[(M * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

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

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

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

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

      7. lower-neg.f64 N/A

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

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

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

      9. lower-neg.f64 36.4%

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

   7. Applied rewrites 36.4%

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

   if -4.999999999999985e-310 < h < 2e53

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 36.4%

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

   7. Applied rewrites 36.4%

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

   if 2e53 < h

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. sqrt-div N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-unsound-sqrt.f64 32.5%

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

   9. Applied rewrites 32.5%

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

Alternative 4: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{D}{d + d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := M \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(t\_2 \cdot h\right)}{\ell}, \frac{M \cdot D}{-2 \cdot d}, 1\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(1 - \left(t\_0 \cdot M\right) \cdot \frac{\left(h \cdot t\_2\right) \cdot 0.5}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d d)))
        (t_1
         (*
          (* (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)))))
        (t_2 (* M t_0)))
   (if (<= t_1 2e+240)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (fma (/ (* 0.5 (* t_2 h)) l) (/ (* M D) (* -2.0 d)) 1.0))
     (if (<= t_1 INFINITY)
       (/ (/ (* (sqrt (* d h)) (sqrt (* d l))) l) h)
       (*
        (sqrt (* (/ d (* h l)) d))
        (- 1.0 (* (* t_0 M) (/ (* (* h t_2) 0.5) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = D / (d + d);
	double t_1 = (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)));
	double t_2 = M * t_0;
	double tmp;
	if (t_1 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * fma(((0.5 * (t_2 * h)) / l), ((M * D) / (-2.0 * d)), 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (1.0 - ((t_0 * M) * (((h * t_2) * 0.5) / l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D / Float64(d + d))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(M * t_0)
	tmp = 0.0
	if (t_1 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * fma(Float64(Float64(0.5 * Float64(t_2 * h)) / l), Float64(Float64(M * D) / Float64(-2.0 * d)), 1.0));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / l) / h);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(1.0 - Float64(Float64(t_0 * M) * Float64(Float64(Float64(h * t_2) * 0.5) / l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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]}, Block[{t$95$2 = N[(M * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * M), $MachinePrecision] * N[(N[(N[(h * t$95$2), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 70.2%

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

   if 2.00000000000000003e240 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.9%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6%

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 32.2%

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

   7. Applied rewrites 32.2%

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

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

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-sqrt.f64 56.4%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 56.4%

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

   9. Applied rewrites 56.4%

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

Alternative 5: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_0}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{t\_0}{d + d}\\ t_4 := t\_3 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - t\_4 \cdot \frac{0.25 \cdot \frac{t\_0 \cdot \left(t\_1 \cdot h\right)}{d}}{\ell}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(1 - t\_4 \cdot \frac{\left(h \cdot \left(t\_1 \cdot t\_3\right)\right) \cdot 0.5}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_1 t_0) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (/ t_0 (+ d d)))
        (t_4 (* t_3 t_1)))
   (if (<= t_2 2e+240)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* t_4 (/ (* 0.25 (/ (* t_0 (* t_1 h)) d)) l))))
     (if (<= t_2 INFINITY)
       (/ (/ (* (sqrt (* d h)) (sqrt (* d l))) l) h)
       (*
        (sqrt (* (/ d (* h l)) d))
        (- 1.0 (* t_4 (/ (* (* h (* t_1 t_3)) 0.5) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_0) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_0 / (d + d);
	double t_4 = t_3 * t_1;
	double tmp;
	if (t_2 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_4 * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	} else {
		tmp = sqrt(((d / (h * l)) * d)) * (1.0 - (t_4 * (((h * (t_1 * t_3)) * 0.5) / l)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_1 * t_0) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_0 / (d + d);
	double t_4 = t_3 * t_1;
	double tmp;
	if (t_2 <= 2e+240) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (t_4 * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.sqrt((d * h)) * Math.sqrt((d * l))) / l) / h;
	} else {
		tmp = Math.sqrt(((d / (h * l)) * d)) * (1.0 - (t_4 * (((h * (t_1 * t_3)) * 0.5) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_1 * t_0) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = t_0 / (d + d)
	t_4 = t_3 * t_1
	tmp = 0
	if t_2 <= 2e+240:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (t_4 * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)))
	elif t_2 <= math.inf:
		tmp = ((math.sqrt((d * h)) * math.sqrt((d * l))) / l) / h
	else:
		tmp = math.sqrt(((d / (h * l)) * d)) * (1.0 - (t_4 * (((h * (t_1 * t_3)) * 0.5) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_1 * t_0) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(t_0 / Float64(d + d))
	t_4 = Float64(t_3 * t_1)
	tmp = 0.0
	if (t_2 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(t_4 * Float64(Float64(0.25 * Float64(Float64(t_0 * Float64(t_1 * h)) / d)) / l))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / l) / h);
	else
		tmp = Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(1.0 - Float64(t_4 * Float64(Float64(Float64(h * Float64(t_1 * t_3)) * 0.5) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_1 * t_0) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = t_0 / (d + d);
	t_4 = t_3 * t_1;
	tmp = 0.0;
	if (t_2 <= 2e+240)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_4 * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	elseif (t_2 <= Inf)
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	else
		tmp = sqrt(((d / (h * l)) * d)) * (1.0 - (t_4 * (((h * (t_1 * t_3)) * 0.5) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$4 * N[(N[(0.25 * N[(N[(t$95$0 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision], N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$4 * N[(N[(N[(h * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 66.3%

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

   10. Applied rewrites 66.3%

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

   if 2.00000000000000003e240 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.9%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6%

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 32.2%

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

   7. Applied rewrites 32.2%

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

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

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-sqrt.f64 56.4%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 56.4%

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

   9. Applied rewrites 56.4%

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

Alternative 6: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq 10^{+186}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_2\right) \cdot \left(1 - \left(\frac{t\_0}{d + d} \cdot t\_1\right) \cdot \frac{0.25 \cdot \frac{t\_0 \cdot \left(t\_1 \cdot h\right)}{d}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot t\_2}{\ell \cdot \sqrt{\frac{1}{\ell}}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (sqrt (/ d h))))
   (if (<= l 1e+186)
     (*
      (* (sqrt (/ d l)) t_2)
      (-
       1.0
       (* (* (/ t_0 (+ d d)) t_1) (/ (* 0.25 (/ (* t_0 (* t_1 h)) d)) l))))
     (/ (* (sqrt d) t_2) (* l (sqrt (/ 1.0 l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = sqrt((d / h));
	double tmp;
	if (l <= 1e+186) {
		tmp = (sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else {
		tmp = (sqrt(d) * t_2) / (l * sqrt((1.0 / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(m), abs(d_1))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = sqrt((d / h))
    if (l <= 1d+186) then
        tmp = (sqrt((d / l)) * t_2) * (1.0d0 - (((t_0 / (d + d)) * t_1) * ((0.25d0 * ((t_0 * (t_1 * h)) / d)) / l)))
    else
        tmp = (sqrt(d) * t_2) / (l * sqrt((1.0d0 / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (l <= 1e+186) {
		tmp = (Math.sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else {
		tmp = (Math.sqrt(d) * t_2) / (l * Math.sqrt((1.0 / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if l <= 1e+186:
		tmp = (math.sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)))
	else:
		tmp = (math.sqrt(d) * t_2) / (l * math.sqrt((1.0 / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= 1e+186)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_2) * Float64(1.0 - Float64(Float64(Float64(t_0 / Float64(d + d)) * t_1) * Float64(Float64(0.25 * Float64(Float64(t_0 * Float64(t_1 * h)) / d)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) * t_2) / Float64(l * sqrt(Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (l <= 1e+186)
		tmp = (sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * ((0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	else
		tmp = (sqrt(d) * t_2) / (l * sqrt((1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1e+186], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(0.25 * N[(N[(t$95$0 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * t$95$2), $MachinePrecision] / N[(l * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if l < 9.9999999999999998e185

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 65.5%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 65.5%

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

   5. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 66.3%

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

   10. Applied rewrites 66.3%

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

   if 9.9999999999999998e185 < l

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \color{blue}{\left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 53.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-/.f64 N/A

          \[\leadsto {\left(d \cdot \color{blue}{\frac{d}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h \cdot \ell}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h \cdot \ell}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. lower-unsound-sqrt.f64 26.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell \cdot h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell \cdot h}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{d}{h}}{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{\color{blue}{\frac{d}{h}}}{\ell}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-unsound-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-unsound-sqrt.f64 28.7%

          \[\leadsto \left(\frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Applied rewrites 28.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\color{blue}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\color{blue}{\ell} \cdot \sqrt{\frac{1}{\ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \color{blue}{\sqrt{\frac{1}{\ell}}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      8. lower-/.f64 22.8%

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

   10. Applied rewrites 22.8%

       \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_4 := t\_1 \cdot t\_3\\ t_5 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_4}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_6 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(t\_4 \cdot t\_1\right) \cdot t\_3\right) \cdot h}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}, -0.5, 1\right) \cdot t\_6\right) \cdot t\_2\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{t\_0}{h \cdot \ell}\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(t\_2 \cdot t\_6\right) \cdot 1\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\frac{t\_0}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_4 \cdot \left(t\_4 \cdot h\right)}{\left(d + d\right) \cdot \left(\left(d + d\right) \cdot \ell\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (* d h)) (sqrt (* d l))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (sqrt (/ d l)))
        (t_3 (fmax (fabs M) (fabs D)))
        (t_4 (* t_1 t_3))
        (t_5
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_4 (* 2.0 d)) 2.0)) (/ h l)))))
        (t_6 (sqrt (/ d h))))
   (if (<= t_5 -2e-7)
     (*
      (*
       (fma (/ (* (* (* t_4 t_1) t_3) h) (* (* (* d d) 4.0) l)) -0.5 1.0)
       t_6)
      t_2)
     (if (<= t_5 0.0)
       (/ t_0 (* h l))
       (if (<= t_5 2e+240)
         (* (* t_2 t_6) 1.0)
         (if (<= t_5 INFINITY)
           (/ (/ t_0 l) h)
           (*
            (fma (/ (* t_4 (* t_4 h)) (* (+ d d) (* (+ d d) l))) -0.5 1.0)
            (sqrt (* (/ d (* l h)) d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d * h)) * sqrt((d * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = sqrt((d / l));
	double t_3 = fmax(fabs(M), fabs(D));
	double t_4 = t_1 * t_3;
	double t_5 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_4 / (2.0 * d)), 2.0)) * (h / l)));
	double t_6 = sqrt((d / h));
	double tmp;
	if (t_5 <= -2e-7) {
		tmp = (fma(((((t_4 * t_1) * t_3) * h) / (((d * d) * 4.0) * l)), -0.5, 1.0) * t_6) * t_2;
	} else if (t_5 <= 0.0) {
		tmp = t_0 / (h * l);
	} else if (t_5 <= 2e+240) {
		tmp = (t_2 * t_6) * 1.0;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (t_0 / l) / h;
	} else {
		tmp = fma(((t_4 * (t_4 * h)) / ((d + d) * ((d + d) * l))), -0.5, 1.0) * sqrt(((d / (l * h)) * d));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = sqrt(Float64(d / l))
	t_3 = fmax(abs(M), abs(D))
	t_4 = Float64(t_1 * t_3)
	t_5 = 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(t_4 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_6 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_5 <= -2e-7)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(t_4 * t_1) * t_3) * h) / Float64(Float64(Float64(d * d) * 4.0) * l)), -0.5, 1.0) * t_6) * t_2);
	elseif (t_5 <= 0.0)
		tmp = Float64(t_0 / Float64(h * l));
	elseif (t_5 <= 2e+240)
		tmp = Float64(Float64(t_2 * t_6) * 1.0);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(t_0 / l) / h);
	else
		tmp = Float64(fma(Float64(Float64(t_4 * Float64(t_4 * h)) / Float64(Float64(d + d) * Float64(Float64(d + d) * l))), -0.5, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = 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[(t$95$4 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, -2e-7], N[(N[(N[(N[(N[(N[(N[(t$95$4 * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision] * h), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(t$95$0 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2e+240], N[(N[(t$95$2 * t$95$6), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(t$95$0 / l), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(t$95$4 * N[(t$95$4 * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 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)))) < -1.9999999999999999e-7

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   7. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot D\right) \cdot M\right) \cdot D\right) \cdot h}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -1.9999999999999999e-7 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 2.00000000000000003e240 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.2%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.2%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\left(d + d\right) \cdot \left(\left(d + d\right) \cdot \ell\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 65.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq 10^{+186}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_2\right) \cdot \left(1 - \left(\frac{t\_0}{d + d} \cdot t\_1\right) \cdot \left(0.25 \cdot \frac{t\_0 \cdot \left(t\_1 \cdot h\right)}{d \cdot \ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot t\_2}{\ell \cdot \sqrt{\frac{1}{\ell}}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmax (fabs M) (fabs D)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (sqrt (/ d h))))
   (if (<= l 1e+186)
     (*
      (* (sqrt (/ d l)) t_2)
      (-
       1.0
       (* (* (/ t_0 (+ d d)) t_1) (* 0.25 (/ (* t_0 (* t_1 h)) (* d l))))))
     (/ (* (sqrt d) t_2) (* l (sqrt (/ 1.0 l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = sqrt((d / h));
	double tmp;
	if (l <= 1e+186) {
		tmp = (sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * (0.25 * ((t_0 * (t_1 * h)) / (d * l)))));
	} else {
		tmp = (sqrt(d) * t_2) / (l * sqrt((1.0 / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(m), abs(d_1))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = sqrt((d / h))
    if (l <= 1d+186) then
        tmp = (sqrt((d / l)) * t_2) * (1.0d0 - (((t_0 / (d + d)) * t_1) * (0.25d0 * ((t_0 * (t_1 * h)) / (d * l)))))
    else
        tmp = (sqrt(d) * t_2) / (l * sqrt((1.0d0 / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (l <= 1e+186) {
		tmp = (Math.sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * (0.25 * ((t_0 * (t_1 * h)) / (d * l)))));
	} else {
		tmp = (Math.sqrt(d) * t_2) / (l * Math.sqrt((1.0 / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if l <= 1e+186:
		tmp = (math.sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * (0.25 * ((t_0 * (t_1 * h)) / (d * l)))))
	else:
		tmp = (math.sqrt(d) * t_2) / (l * math.sqrt((1.0 / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= 1e+186)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_2) * Float64(1.0 - Float64(Float64(Float64(t_0 / Float64(d + d)) * t_1) * Float64(0.25 * Float64(Float64(t_0 * Float64(t_1 * h)) / Float64(d * l))))));
	else
		tmp = Float64(Float64(sqrt(d) * t_2) / Float64(l * sqrt(Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (l <= 1e+186)
		tmp = (sqrt((d / l)) * t_2) * (1.0 - (((t_0 / (d + d)) * t_1) * (0.25 * ((t_0 * (t_1 * h)) / (d * l)))));
	else
		tmp = (sqrt(d) * t_2) / (l * sqrt((1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1e+186], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.25 * N[(N[(t$95$0 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * t$95$2), $MachinePrecision] / N[(l * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if l < 9.9999999999999998e185

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d} \cdot \ell}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)\right) \]

      5. lower-*.f64 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\ell}}\right)\right) \]

   8. Applied rewrites 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]

   if 9.9999999999999998e185 < l

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \color{blue}{\left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 53.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-/.f64 N/A

          \[\leadsto {\left(d \cdot \color{blue}{\frac{d}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h \cdot \ell}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h \cdot \ell}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. lower-unsound-sqrt.f64 26.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell \cdot h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell \cdot h}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{d}{h}}{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{\color{blue}{\frac{d}{h}}}{\ell}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-unsound-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-unsound-sqrt.f64 28.7%

          \[\leadsto \left(\frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Applied rewrites 28.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\color{blue}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\color{blue}{\ell} \cdot \sqrt{\frac{1}{\ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \color{blue}{\sqrt{\frac{1}{\ell}}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      8. lower-/.f64 22.8%

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

   10. Applied rewrites 22.8%

       \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_4 := t\_1 \cdot t\_3\\ t_5 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_4}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_6 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\left(t\_4 \cdot t\_1\right) \cdot t\_3\right) \cdot h}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}, -0.5, 1\right) \cdot t\_6\right) \cdot t\_2\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{t\_0}{h \cdot \ell}\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(t\_2 \cdot t\_6\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\ell}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (* d h)) (sqrt (* d l))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (sqrt (/ d l)))
        (t_3 (fmax (fabs M) (fabs D)))
        (t_4 (* t_1 t_3))
        (t_5
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_4 (* 2.0 d)) 2.0)) (/ h l)))))
        (t_6 (sqrt (/ d h))))
   (if (<= t_5 -2e-7)
     (*
      (*
       (fma (/ (* (* (* t_4 t_1) t_3) h) (* (* (* d d) 4.0) l)) -0.5 1.0)
       t_6)
      t_2)
     (if (<= t_5 0.0)
       (/ t_0 (* h l))
       (if (<= t_5 2e+240) (* (* t_2 t_6) 1.0) (/ (/ t_0 l) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d * h)) * sqrt((d * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = sqrt((d / l));
	double t_3 = fmax(fabs(M), fabs(D));
	double t_4 = t_1 * t_3;
	double t_5 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_4 / (2.0 * d)), 2.0)) * (h / l)));
	double t_6 = sqrt((d / h));
	double tmp;
	if (t_5 <= -2e-7) {
		tmp = (fma(((((t_4 * t_1) * t_3) * h) / (((d * d) * 4.0) * l)), -0.5, 1.0) * t_6) * t_2;
	} else if (t_5 <= 0.0) {
		tmp = t_0 / (h * l);
	} else if (t_5 <= 2e+240) {
		tmp = (t_2 * t_6) * 1.0;
	} else {
		tmp = (t_0 / l) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = sqrt(Float64(d / l))
	t_3 = fmax(abs(M), abs(D))
	t_4 = Float64(t_1 * t_3)
	t_5 = 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(t_4 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_6 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_5 <= -2e-7)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(t_4 * t_1) * t_3) * h) / Float64(Float64(Float64(d * d) * 4.0) * l)), -0.5, 1.0) * t_6) * t_2);
	elseif (t_5 <= 0.0)
		tmp = Float64(t_0 / Float64(h * l));
	elseif (t_5 <= 2e+240)
		tmp = Float64(Float64(t_2 * t_6) * 1.0);
	else
		tmp = Float64(Float64(t_0 / l) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = 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[(t$95$4 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, -2e-7], N[(N[(N[(N[(N[(N[(N[(t$95$4 * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision] * h), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(t$95$0 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2e+240], N[(N[(t$95$2 * t$95$6), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$0 / l), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 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)))) < -1.9999999999999999e-7

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   7. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot D\right) \cdot M\right) \cdot D\right) \cdot h}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -1.9999999999999999e-7 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 2.00000000000000003e240 < (*.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 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.2%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.2%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\left(\left(t\_2 \cdot t\_2\right) \cdot h\right) \cdot 0.25\right) \cdot \left(t\_1 \cdot t\_1\right)}{\ell \cdot d}}{d}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{t\_0}{h \cdot \ell}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\ell}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (* d h)) (sqrt (* d l))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_1 t_2) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_3 -2e-7)
     (*
      (fma
       (/ (/ (* (* (* (* t_2 t_2) h) 0.25) (* t_1 t_1)) (* l d)) d)
       -0.5
       1.0)
      (sqrt (* d (/ d (* h l)))))
     (if (<= t_3 0.0)
       (/ t_0 (* h l))
       (if (<= t_3 2e+240)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (/ (/ t_0 l) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d * h)) * sqrt((d * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= -2e-7) {
		tmp = fma(((((((t_2 * t_2) * h) * 0.25) * (t_1 * t_1)) / (l * d)) / d), -0.5, 1.0) * sqrt((d * (d / (h * l))));
	} else if (t_3 <= 0.0) {
		tmp = t_0 / (h * l);
	} else if (t_3 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (t_0 / l) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = 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(t_1 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= -2e-7)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(t_2 * t_2) * h) * 0.25) * Float64(t_1 * t_1)) / Float64(l * d)) / d), -0.5, 1.0) * sqrt(Float64(d * Float64(d / Float64(h * l)))));
	elseif (t_3 <= 0.0)
		tmp = Float64(t_0 / Float64(h * l));
	elseif (t_3 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(t_0 / l) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$1 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-7], N[(N[(N[(N[(N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * h), $MachinePrecision] * 0.25), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(t$95$0 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$0 / l), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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)))) < -1.9999999999999999e-7

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot d}}{d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot d}}{d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot d}}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(M \cdot M\right)\right)}}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\color{blue}{\left(\left(h \cdot \left(D \cdot D\right)\right) \cdot \frac{1}{4}\right)} \cdot \left(M \cdot M\right)}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\ell \cdot d}}{d}, \frac{-1}{2}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      18. lower-*.f64 42.1%

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot d}}}{d}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 42.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\ell \cdot d}}{d}}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   if -1.9999999999999999e-7 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 2.00000000000000003e240 < (*.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 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.2%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.2%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(t\_3 \cdot t\_1\right) \cdot t\_2\right) \cdot h}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{t\_0}{h \cdot \ell}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\ell}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (* d h)) (sqrt (* d l))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3 (* t_1 t_2))
        (t_4
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_3 (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_4 -2e-7)
     (*
      (fma (/ (* (* (* t_3 t_1) t_2) h) (* (* (* d d) 4.0) l)) -0.5 1.0)
      (sqrt (* (/ d (* h l)) d)))
     (if (<= t_4 0.0)
       (/ t_0 (* h l))
       (if (<= t_4 2e+240)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (/ (/ t_0 l) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d * h)) * sqrt((d * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = t_1 * t_2;
	double t_4 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_3 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_4 <= -2e-7) {
		tmp = fma(((((t_3 * t_1) * t_2) * h) / (((d * d) * 4.0) * l)), -0.5, 1.0) * sqrt(((d / (h * l)) * d));
	} else if (t_4 <= 0.0) {
		tmp = t_0 / (h * l);
	} else if (t_4 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (t_0 / l) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(t_1 * t_2)
	t_4 = 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(t_3 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_4 <= -2e-7)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(t_3 * t_1) * t_2) * h) / Float64(Float64(Float64(d * d) * 4.0) * l)), -0.5, 1.0) * sqrt(Float64(Float64(d / Float64(h * l)) * d)));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_0 / Float64(h * l));
	elseif (t_4 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(t_0 / l) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = 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[(t$95$3 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-7], N[(N[(N[(N[(N[(N[(t$95$3 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] * h), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$0 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$0 / l), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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)))) < -1.9999999999999999e-7

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   7. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot D\right) \cdot M\right) \cdot D\right) \cdot h}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}} \]

   if -1.9999999999999999e-7 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 2.00000000000000003e240 < (*.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 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.2%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.2%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(t\_2 \cdot t\_2\right) \cdot t\_1\right) \cdot t\_1\right) \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{t\_0}{h \cdot \ell}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\ell}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (* d h)) (sqrt (* d l))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_1 t_2) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_3 -2e-7)
     (*
      (fma
       (* (* (* (* (* t_2 t_2) t_1) t_1) (/ 0.25 (* (* d d) l))) h)
       -0.5
       1.0)
      (sqrt (* (/ d (* h l)) d)))
     (if (<= t_3 0.0)
       (/ t_0 (* h l))
       (if (<= t_3 2e+240)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (/ (/ t_0 l) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d * h)) * sqrt((d * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= -2e-7) {
		tmp = fma((((((t_2 * t_2) * t_1) * t_1) * (0.25 / ((d * d) * l))) * h), -0.5, 1.0) * sqrt(((d / (h * l)) * d));
	} else if (t_3 <= 0.0) {
		tmp = t_0 / (h * l);
	} else if (t_3 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (t_0 / l) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = 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(t_1 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= -2e-7)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(t_2 * t_2) * t_1) * t_1) * Float64(0.25 / Float64(Float64(d * d) * l))) * h), -0.5, 1.0) * sqrt(Float64(Float64(d / Float64(h * l)) * d)));
	elseif (t_3 <= 0.0)
		tmp = Float64(t_0 / Float64(h * l));
	elseif (t_3 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(t_0 / l) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$1 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-7], N[(N[(N[(N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $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[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(t$95$0 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$0 / l), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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)))) < -1.9999999999999999e-7

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot \frac{0.25}{\left(d \cdot d\right) \cdot \ell}\right) \cdot h, -0.5, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}} \]

   if -1.9999999999999999e-7 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 2.00000000000000003e240 < (*.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 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.2%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.2%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{t\_1}{h \cdot \ell}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{\ell}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (* d h)) (sqrt (* d l)))))
   (if (<= t_0 -5e-34)
     (* -1.0 (sqrt (/ (pow d 2.0) (* h l))))
     (if (<= t_0 0.0)
       (/ t_1 (* h l))
       (if (<= t_0 2e+240)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (/ (/ t_1 l) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d * h)) * sqrt((d * l));
	double tmp;
	if (t_0 <= -5e-34) {
		tmp = -1.0 * sqrt((pow(d, 2.0) / (h * l)));
	} else if (t_0 <= 0.0) {
		tmp = t_1 / (h * l);
	} else if (t_0 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (t_1 / l) / h;
	}
	return tmp;
}

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
    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 * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d * h)) * sqrt((d * l))
    if (t_0 <= (-5d-34)) then
        tmp = (-1.0d0) * sqrt(((d ** 2.0d0) / (h * l)))
    else if (t_0 <= 0.0d0) then
        tmp = t_1 / (h * l)
    else if (t_0 <= 2d+240) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (t_1 / l) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = Math.sqrt((d * h)) * Math.sqrt((d * l));
	double tmp;
	if (t_0 <= -5e-34) {
		tmp = -1.0 * Math.sqrt((Math.pow(d, 2.0) / (h * l)));
	} else if (t_0 <= 0.0) {
		tmp = t_1 / (h * l);
	} else if (t_0 <= 2e+240) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (t_1 / l) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d * h)) * math.sqrt((d * l))
	tmp = 0
	if t_0 <= -5e-34:
		tmp = -1.0 * math.sqrt((math.pow(d, 2.0) / (h * l)))
	elif t_0 <= 0.0:
		tmp = t_1 / (h * l)
	elif t_0 <= 2e+240:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (t_1 / l) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	tmp = 0.0
	if (t_0 <= -5e-34)
		tmp = Float64(-1.0 * sqrt(Float64((d ^ 2.0) / Float64(h * l))));
	elseif (t_0 <= 0.0)
		tmp = Float64(t_1 / Float64(h * l));
	elseif (t_0 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(t_1 / l) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d * h)) * sqrt((d * l));
	tmp = 0.0;
	if (t_0 <= -5e-34)
		tmp = -1.0 * sqrt(((d ^ 2.0) / (h * l)));
	elseif (t_0 <= 0.0)
		tmp = t_1 / (h * l);
	elseif (t_0 <= 2e+240)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = (t_1 / l) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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]}, Block[{t$95$1 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-34], N[(-1.0 * N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(t$95$1 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$1 / l), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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)))) < -5.0000000000000003e-34

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.7%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      4. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      5. lower-*.f64 11.9%

         \[\leadsto -1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

   9. Applied rewrites 11.9%

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -5.0000000000000003e-34 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 2.00000000000000003e240 < (*.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 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.2%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.2%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-h\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{t\_1}{h \cdot \ell}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{\ell}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (* d h)) (sqrt (* d l)))))
   (if (<= t_0 -5e-34)
     (/ (* (sqrt (* (/ d (* h l)) d)) (- h)) h)
     (if (<= t_0 0.0)
       (/ t_1 (* h l))
       (if (<= t_0 2e+240)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (/ (/ t_1 l) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d * h)) * sqrt((d * l));
	double tmp;
	if (t_0 <= -5e-34) {
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1 / (h * l);
	} else if (t_0 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (t_1 / l) / h;
	}
	return tmp;
}

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
    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 * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d * h)) * sqrt((d * l))
    if (t_0 <= (-5d-34)) then
        tmp = (sqrt(((d / (h * l)) * d)) * -h) / h
    else if (t_0 <= 0.0d0) then
        tmp = t_1 / (h * l)
    else if (t_0 <= 2d+240) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (t_1 / l) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = Math.sqrt((d * h)) * Math.sqrt((d * l));
	double tmp;
	if (t_0 <= -5e-34) {
		tmp = (Math.sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1 / (h * l);
	} else if (t_0 <= 2e+240) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (t_1 / l) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d * h)) * math.sqrt((d * l))
	tmp = 0
	if t_0 <= -5e-34:
		tmp = (math.sqrt(((d / (h * l)) * d)) * -h) / h
	elif t_0 <= 0.0:
		tmp = t_1 / (h * l)
	elif t_0 <= 2e+240:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (t_1 / l) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	tmp = 0.0
	if (t_0 <= -5e-34)
		tmp = Float64(Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(-h)) / h);
	elseif (t_0 <= 0.0)
		tmp = Float64(t_1 / Float64(h * l));
	elseif (t_0 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(t_1 / l) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d * h)) * sqrt((d * l));
	tmp = 0.0;
	if (t_0 <= -5e-34)
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	elseif (t_0 <= 0.0)
		tmp = t_1 / (h * l);
	elseif (t_0 <= 2e+240)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = (t_1 / l) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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]}, Block[{t$95$1 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-34], N[(N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * (-h)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(t$95$1 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$1 / l), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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)))) < -5.0000000000000003e-34

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot h\right)}{h} \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\mathsf{neg}\left(h\right)\right)}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\mathsf{neg}\left(h\right)\right)}{h} \]

   9. Applied rewrites 13.7%

      \[\leadsto \frac{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-h\right)}{h} \]

   if -5.0000000000000003e-34 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 2.00000000000000003e240 < (*.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 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 32.2%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 32.2%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-h\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (* (sqrt (* d h)) (sqrt (* d l))) (* h l))))
   (if (<= t_0 -5e-34)
     (/ (* (sqrt (* (/ d (* h l)) d)) (- h)) h)
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 2e+240) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	double tmp;
	if (t_0 <= -5e-34) {
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+240) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    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 * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (sqrt((d * h)) * sqrt((d * l))) / (h * l)
    if (t_0 <= (-5d-34)) then
        tmp = (sqrt(((d / (h * l)) * d)) * -h) / h
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+240) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.sqrt((d * h)) * Math.sqrt((d * l))) / (h * l);
	double tmp;
	if (t_0 <= -5e-34) {
		tmp = (Math.sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+240) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.sqrt((d * h)) * math.sqrt((d * l))) / (h * l)
	tmp = 0
	if t_0 <= -5e-34:
		tmp = (math.sqrt(((d / (h * l)) * d)) * -h) / h
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+240:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / Float64(h * l))
	tmp = 0.0
	if (t_0 <= -5e-34)
		tmp = Float64(Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(-h)) / h);
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+240)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	tmp = 0.0;
	if (t_0 <= -5e-34)
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+240)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-34], N[(N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * (-h)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+240], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]]

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)))) < -5.0000000000000003e-34

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot h\right)}{h} \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\mathsf{neg}\left(h\right)\right)}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\mathsf{neg}\left(h\right)\right)}{h} \]

   9. Applied rewrites 13.7%

      \[\leadsto \frac{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-h\right)}{h} \]

   if -5.0000000000000003e-34 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.00000000000000003e240 < (*.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 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

      7. lower-/.f64 9.7%

         \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-1 \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{h} \]

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 30.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   10. Applied rewrites 30.0%

       \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000003e240

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{+208}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_0\right) \cdot 1\\ \mathbf{elif}\;h \leq 1.25 \cdot 10^{-287}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot t\_0}{\ell \cdot \sqrt{\frac{1}{\ell}}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= h -5e+208)
     (* (* (sqrt (/ d l)) t_0) 1.0)
     (if (<= h 1.25e-287)
       (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
       (/ (* (sqrt d) t_0) (* l (sqrt (/ 1.0 l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (h <= -5e+208) {
		tmp = (sqrt((d / l)) * t_0) * 1.0;
	} else if (h <= 1.25e-287) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (sqrt(d) * t_0) / (l * sqrt((1.0 / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / h))
    if (h <= (-5d+208)) then
        tmp = (sqrt((d / l)) * t_0) * 1.0d0
    else if (h <= 1.25d-287) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (sqrt(d) * t_0) / (l * sqrt((1.0d0 / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double tmp;
	if (h <= -5e+208) {
		tmp = (Math.sqrt((d / l)) * t_0) * 1.0;
	} else if (h <= 1.25e-287) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (Math.sqrt(d) * t_0) / (l * Math.sqrt((1.0 / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	tmp = 0
	if h <= -5e+208:
		tmp = (math.sqrt((d / l)) * t_0) * 1.0
	elif h <= 1.25e-287:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (math.sqrt(d) * t_0) / (l * math.sqrt((1.0 / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (h <= -5e+208)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_0) * 1.0);
	elseif (h <= 1.25e-287)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(sqrt(d) * t_0) / Float64(l * sqrt(Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	tmp = 0.0;
	if (h <= -5e+208)
		tmp = (sqrt((d / l)) * t_0) * 1.0;
	elseif (h <= 1.25e-287)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (sqrt(d) * t_0) / (l * sqrt((1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e+208], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[h, 1.25e-287], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(l * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.0000000000000004e208

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if -5.0000000000000004e208 < h < 1.25000000000000006e-287

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. 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 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 1.25000000000000006e-287 < h

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \color{blue}{\left(\left(\frac{D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 53.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto {\color{blue}{\left(d \cdot \frac{d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-/.f64 N/A

          \[\leadsto {\left(d \cdot \color{blue}{\frac{d}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h \cdot \ell}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h \cdot \ell}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. lower-unsound-sqrt.f64 26.4%

          \[\leadsto \left(\sqrt{\frac{d}{\ell \cdot h}} \cdot \color{blue}{\sqrt{d}}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell \cdot h}} \cdot \sqrt{d}\right)} \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\ell \cdot h}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{h \cdot \ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{d}{h}}{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{\color{blue}{\frac{d}{h}}}{\ell}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\ell}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-unsound-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-unsound-sqrt.f64 28.7%

          \[\leadsto \left(\frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Applied rewrites 28.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \cdot \left(1 - M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\color{blue}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\color{blue}{\ell} \cdot \sqrt{\frac{1}{\ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \color{blue}{\sqrt{\frac{1}{\ell}}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

      8. lower-/.f64 22.8%

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}} \]

   10. Applied rewrites 22.8%

       \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\ell \cdot \sqrt{\frac{1}{\ell}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{+208}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;h \leq 1.25 \cdot 10^{-287}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -5e+208)
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
   (if (<= h 1.25e-287)
     (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
     (/ (* d (/ (sqrt h) (sqrt l))) h))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -5e+208) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (h <= 1.25e-287) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (h <= (-5d+208)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else if (h <= 1.25d-287) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -5e+208) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (h <= 1.25e-287) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -5e+208:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif h <= 1.25e-287:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -5e+208)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (h <= 1.25e-287)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -5e+208)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (h <= 1.25e-287)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -5e+208], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[h, 1.25e-287], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.0000000000000004e208

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} \cdot \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      17. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \frac{1}{2}\right)}\right) \]

   3. Applied rewrites 65.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      3. lower-*.f64 65.5%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      8. lift-sqrt.f64 65.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \frac{1}{2}\right)\right) \]

      13. lift-sqrt.f64 65.5%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot 0.5\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if -5.0000000000000004e208 < h < 1.25000000000000006e-287

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. 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 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 1.25000000000000006e-287 < h

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.7%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.2%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.2%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.7%

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.7%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;h \leq 1.25 \cdot 10^{-287}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 1.25e-287)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (/ (* d (/ (sqrt h) (sqrt l))) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.25e-287) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (h <= 1.25d-287) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.25e-287) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 1.25e-287:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 1.25e-287)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 1.25e-287)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 1.25e-287], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.25000000000000006e-287

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. 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 26.3%

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 1.25000000000000006e-287 < h

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.7%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.2%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.2%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.7%

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.7%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 40.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 2.4e-302)
   (/ (* d (sqrt (/ h l))) h)
   (/ (* d (/ (sqrt h) (sqrt l))) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 2.4e-302) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (h <= 2.4d-302) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 2.4e-302) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 2.4e-302:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 2.4e-302)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 2.4e-302)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 2.4e-302], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 2.40000000000000022e-302

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.7%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.2%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.2%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 2.40000000000000022e-302 < h

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.7%

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.7%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 37.2%

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 37.2%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.7%

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.7%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 37.2% accurate, 7.4× speedup

Math

\[\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ (* d (sqrt (/ h l))) h))

C

double code(double d, double h, double l, double M, double D) {
	return (d * sqrt((h / l))) / h;
}

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 * sqrt((h / l))) / h
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (d * Math.sqrt((h / l))) / h;
}

Python

def code(d, h, l, M, D):
	return (d * math.sqrt((h / l))) / h

Julia

function code(d, h, l, M, D)
	return Float64(Float64(d * sqrt(Float64(h / l))) / h)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (d * sqrt((h / l))) / h;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]

Derivation

1. Initial program 66.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in h around 0

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   6. lower-/.f64 23.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

4. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. sqrt-unprod N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   6. lower-*.f64 21.7%

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   8. *-commutative N/A

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   9. lower-*.f64 21.7%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

6. Applied rewrites 21.7%

   \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

7. Taylor expanded in d around 0

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   3. lower-/.f64 37.2%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

9. Applied rewrites 37.2%

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025185 
(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)))))