Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.1% → 84.0%

Time: 11.0s

Alternatives: 25

Speedup: 1.8×

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: 67.1% 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: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := \frac{D}{d + d}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0\\ t_3 := \ell \cdot \left(d + d\right)\\ t_4 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_5 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+200}:\\ \;\;\;\;t\_4 \cdot \left(1 - \left(t\_1 \cdot M\right) \cdot \left(\left(t\_1 \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_5 \cdot \left(1 - \frac{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{t\_3}\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;t\_4 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \left(1 - \frac{\left(\frac{\left(M \cdot D\right) \cdot 0.25}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{t\_3}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_1 (/ D (+ d d)))
        (t_2 (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) t_0))
        (t_3 (* l (+ d d)))
        (t_4 (* (sqrt (/ d l)) (sqrt (/ d h))))
        (t_5 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_2 -5e+200)
     (* t_4 (- 1.0 (* (* t_1 M) (* (* t_1 (* M 0.5)) (/ h l)))))
     (if (<= t_2 0.0)
       (* t_5 (- 1.0 (/ (* (* 0.25 (/ (* D (* M h)) d)) (* M D)) t_3)))
       (if (<= t_2 4e+233)
         (* t_4 t_0)
         (* t_5 (- 1.0 (/ (* (* (/ (* (* M D) 0.25) d) h) (* M D)) t_3))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = D / (d + d);
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * t_0;
	double t_3 = l * (d + d);
	double t_4 = sqrt((d / l)) * sqrt((d / h));
	double t_5 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_2 <= -5e+200) {
		tmp = t_4 * (1.0 - ((t_1 * M) * ((t_1 * (M * 0.5)) * (h / l))));
	} else if (t_2 <= 0.0) {
		tmp = t_5 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3));
	} else if (t_2 <= 4e+233) {
		tmp = t_4 * t_0;
	} else {
		tmp = t_5 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3));
	}
	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) :: t_5
    real(8) :: tmp
    t_0 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    t_1 = d_1 / (d + d)
    t_2 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * t_0
    t_3 = l * (d + d)
    t_4 = sqrt((d / l)) * sqrt((d / h))
    t_5 = abs((d / sqrt((l * h))))
    if (t_2 <= (-5d+200)) then
        tmp = t_4 * (1.0d0 - ((t_1 * m) * ((t_1 * (m * 0.5d0)) * (h / l))))
    else if (t_2 <= 0.0d0) then
        tmp = t_5 * (1.0d0 - (((0.25d0 * ((d_1 * (m * h)) / d)) * (m * d_1)) / t_3))
    else if (t_2 <= 4d+233) then
        tmp = t_4 * t_0
    else
        tmp = t_5 * (1.0d0 - ((((((m * d_1) * 0.25d0) / d) * h) * (m * d_1)) / t_3))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = D / (d + d);
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * t_0;
	double t_3 = l * (d + d);
	double t_4 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double t_5 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_2 <= -5e+200) {
		tmp = t_4 * (1.0 - ((t_1 * M) * ((t_1 * (M * 0.5)) * (h / l))));
	} else if (t_2 <= 0.0) {
		tmp = t_5 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3));
	} else if (t_2 <= 4e+233) {
		tmp = t_4 * t_0;
	} else {
		tmp = t_5 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l))
	t_1 = D / (d + d)
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * t_0
	t_3 = l * (d + d)
	t_4 = math.sqrt((d / l)) * math.sqrt((d / h))
	t_5 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_2 <= -5e+200:
		tmp = t_4 * (1.0 - ((t_1 * M) * ((t_1 * (M * 0.5)) * (h / l))))
	elif t_2 <= 0.0:
		tmp = t_5 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3))
	elif t_2 <= 4e+233:
		tmp = t_4 * t_0
	else:
		tmp = t_5 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3))
	return tmp

Julia

function code(d, h, l, M, D)
	t_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(D / Float64(d + d))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_0)
	t_3 = Float64(l * Float64(d + d))
	t_4 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	t_5 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_2 <= -5e+200)
		tmp = Float64(t_4 * Float64(1.0 - Float64(Float64(t_1 * M) * Float64(Float64(t_1 * Float64(M * 0.5)) * Float64(h / l)))));
	elseif (t_2 <= 0.0)
		tmp = Float64(t_5 * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(Float64(D * Float64(M * h)) / d)) * Float64(M * D)) / t_3)));
	elseif (t_2 <= 4e+233)
		tmp = Float64(t_4 * t_0);
	else
		tmp = Float64(t_5 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) * 0.25) / d) * h) * Float64(M * D)) / t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l));
	t_1 = D / (d + d);
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * t_0;
	t_3 = l * (d + d);
	t_4 = sqrt((d / l)) * sqrt((d / h));
	t_5 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_2 <= -5e+200)
		tmp = t_4 * (1.0 - ((t_1 * M) * ((t_1 * (M * 0.5)) * (h / l))));
	elseif (t_2 <= 0.0)
		tmp = t_5 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3));
	elseif (t_2 <= 4e+233)
		tmp = t_4 * t_0;
	else
		tmp = t_5 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(D / N[(d + 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] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e+200], N[(t$95$4 * N[(1.0 - N[(N[(t$95$1 * M), $MachinePrecision] * N[(N[(t$95$1 * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$5 * N[(1.0 - N[(N[(N[(0.25 * N[(N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+233], N[(t$95$4 * t$95$0), $MachinePrecision], N[(t$95$5 * N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $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)))) < -5.00000000000000019e200

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.0%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 68.0%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 68.0%

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

   5. Applied rewrites 68.0%

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

   if -5.00000000000000019e200 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.4%

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

   10. Applied rewrites 73.4%

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

   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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.1%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 67.1%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 67.1%

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

   3. Applied rewrites 67.1%

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

   if 3.99999999999999989e233 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 75.6%

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

   9. Applied rewrites 75.6%

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

Alternative 2: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{D}{d + d}\\ t_1 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := t\_1 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \ell \cdot \left(d + d\right)\\ t_4 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+200}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(t\_0 \cdot M\right) \cdot \left(\left(t\_0 \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_4 \cdot \left(1 - \frac{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{t\_3}\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;t\_1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(1 - \frac{\left(\frac{\left(M \cdot D\right) \cdot 0.25}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{t\_3}\right)\\ \end{array} \]

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))))
        (t_2
         (*
          t_1
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (* l (+ d d)))
        (t_4 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_2 -5e+200)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* t_0 M) (* (* t_0 (* M 0.5)) (/ h l)))))
     (if (<= t_2 0.0)
       (* t_4 (- 1.0 (/ (* (* 0.25 (/ (* D (* M h)) d)) (* M D)) t_3)))
       (if (<= t_2 4e+233)
         (* t_1 1.0)
         (* t_4 (- 1.0 (/ (* (* (/ (* (* M D) 0.25) d) h) (* M D)) t_3))))))))

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));
	double t_2 = t_1 * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = l * (d + d);
	double t_4 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_2 <= -5e+200) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	} else if (t_2 <= 0.0) {
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3));
	} else if (t_2 <= 4e+233) {
		tmp = t_1 * 1.0;
	} else {
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3));
	}
	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 = d_1 / (d + d)
    t_1 = ((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))
    t_2 = t_1 * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_3 = l * (d + d)
    t_4 = abs((d / sqrt((l * h))))
    if (t_2 <= (-5d+200)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((t_0 * m) * ((t_0 * (m * 0.5d0)) * (h / l))))
    else if (t_2 <= 0.0d0) then
        tmp = t_4 * (1.0d0 - (((0.25d0 * ((d_1 * (m * h)) / d)) * (m * d_1)) / t_3))
    else if (t_2 <= 4d+233) then
        tmp = t_1 * 1.0d0
    else
        tmp = t_4 * (1.0d0 - ((((((m * d_1) * 0.25d0) / d) * h) * (m * d_1)) / t_3))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = D / (d + d);
	double t_1 = Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0));
	double t_2 = t_1 * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = l * (d + d);
	double t_4 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_2 <= -5e+200) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	} else if (t_2 <= 0.0) {
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3));
	} else if (t_2 <= 4e+233) {
		tmp = t_1 * 1.0;
	} else {
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = D / (d + d)
	t_1 = math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))
	t_2 = t_1 * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = l * (d + d)
	t_4 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_2 <= -5e+200:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))))
	elif t_2 <= 0.0:
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3))
	elif t_2 <= 4e+233:
		tmp = t_1 * 1.0
	else:
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D / Float64(d + d))
	t_1 = Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))
	t_2 = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(l * Float64(d + d))
	t_4 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_2 <= -5e+200)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_0 * M) * Float64(Float64(t_0 * Float64(M * 0.5)) * Float64(h / l)))));
	elseif (t_2 <= 0.0)
		tmp = Float64(t_4 * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(Float64(D * Float64(M * h)) / d)) * Float64(M * D)) / t_3)));
	elseif (t_2 <= 4e+233)
		tmp = Float64(t_1 * 1.0);
	else
		tmp = Float64(t_4 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) * 0.25) / d) * h) * Float64(M * D)) / t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = D / (d + d);
	t_1 = ((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0));
	t_2 = t_1 * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = l * (d + d);
	t_4 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_2 <= -5e+200)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	elseif (t_2 <= 0.0)
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_3));
	elseif (t_2 <= 4e+233)
		tmp = t_1 * 1.0;
	else
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_3));
	end
	tmp_2 = 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[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]}, Block[{t$95$2 = N[(t$95$1 * 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$3 = N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e+200], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * M), $MachinePrecision] * N[(N[(t$95$0 * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$4 * N[(1.0 - N[(N[(N[(0.25 * N[(N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+233], N[(t$95$1 * 1.0), $MachinePrecision], N[(t$95$4 * N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $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)))) < -5.00000000000000019e200

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.0%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 68.0%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 68.0%

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

   5. Applied rewrites 68.0%

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

   if -5.00000000000000019e200 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.4%

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

   10. Applied rewrites 73.4%

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

   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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 40.1%

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

   if 3.99999999999999989e233 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 75.6%

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

   9. Applied rewrites 75.6%

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

Alternative 3: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{D}{d + d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \ell \cdot \left(d + d\right)\\ t_3 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(t\_0 \cdot M\right) \cdot \left(\left(t\_0 \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right)\\ t_4 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+200}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_4 \cdot \left(1 - \frac{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{t\_2}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(1 - \frac{\left(\frac{\left(M \cdot D\right) \cdot 0.25}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{t\_2}\right)\\ \end{array} \]

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 (* l (+ d d)))
        (t_3
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (* (* t_0 M) (* (* t_0 (* M 0.5)) (/ h l))))))
        (t_4 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_1 -5e+200)
     t_3
     (if (<= t_1 0.0)
       (* t_4 (- 1.0 (/ (* (* 0.25 (/ (* D (* M h)) d)) (* M D)) t_2)))
       (if (<= t_1 4e+233)
         t_3
         (* t_4 (- 1.0 (/ (* (* (/ (* (* M D) 0.25) d) h) (* M D)) t_2))))))))

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 = l * (d + d);
	double t_3 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	double t_4 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_1 <= -5e+200) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_2));
	} else if (t_1 <= 4e+233) {
		tmp = t_3;
	} else {
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_2));
	}
	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 = d_1 / (d + d)
    t_1 = (((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_2 = l * (d + d)
    t_3 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((t_0 * m) * ((t_0 * (m * 0.5d0)) * (h / l))))
    t_4 = abs((d / sqrt((l * h))))
    if (t_1 <= (-5d+200)) then
        tmp = t_3
    else if (t_1 <= 0.0d0) then
        tmp = t_4 * (1.0d0 - (((0.25d0 * ((d_1 * (m * h)) / d)) * (m * d_1)) / t_2))
    else if (t_1 <= 4d+233) then
        tmp = t_3
    else
        tmp = t_4 * (1.0d0 - ((((((m * d_1) * 0.25d0) / d) * h) * (m * d_1)) / t_2))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = D / (d + d);
	double t_1 = (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_2 = l * (d + d);
	double t_3 = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	double t_4 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_1 <= -5e+200) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_2));
	} else if (t_1 <= 4e+233) {
		tmp = t_3;
	} else {
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_2));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = D / (d + d)
	t_1 = (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_2 = l * (d + d)
	t_3 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))))
	t_4 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_1 <= -5e+200:
		tmp = t_3
	elif t_1 <= 0.0:
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_2))
	elif t_1 <= 4e+233:
		tmp = t_3
	else:
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_2))
	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(l * Float64(d + d))
	t_3 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_0 * M) * Float64(Float64(t_0 * Float64(M * 0.5)) * Float64(h / l)))))
	t_4 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= -5e+200)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = Float64(t_4 * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(Float64(D * Float64(M * h)) / d)) * Float64(M * D)) / t_2)));
	elseif (t_1 <= 4e+233)
		tmp = t_3;
	else
		tmp = Float64(t_4 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) * 0.25) / d) * h) * Float64(M * D)) / t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = D / (d + d);
	t_1 = (((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_2 = l * (d + d);
	t_3 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	t_4 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_1 <= -5e+200)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = t_4 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / t_2));
	elseif (t_1 <= 4e+233)
		tmp = t_3;
	else
		tmp = t_4 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / t_2));
	end
	tmp_2 = 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[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * M), $MachinePrecision] * N[(N[(t$95$0 * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e+200], t$95$3, If[LessEqual[t$95$1, 0.0], N[(t$95$4 * N[(1.0 - N[(N[(N[(0.25 * N[(N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+233], t$95$3, N[(t$95$4 * N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$2), $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)))) < -5.00000000000000019e200 or -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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.0%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 68.0%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 68.0%

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

   5. Applied rewrites 68.0%

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

   if -5.00000000000000019e200 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.4%

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

   10. Applied rewrites 73.4%

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

   if 3.99999999999999989e233 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 75.6%

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

   9. Applied rewrites 75.6%

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

Alternative 4: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ t_1 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_4 := \frac{t\_3}{d}\\ t_5 := t\_1 \cdot t\_3\\ t_6 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_5}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_7 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_6 \leq -1 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(1 - \frac{t\_5 \cdot \left(\left(\frac{t\_5}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right) \cdot t\_2\right) \cdot t\_7\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(\frac{t\_3}{d + d} \cdot t\_1\right) \cdot \left(\left(0.25 \cdot \frac{t\_3 \cdot t\_1}{d}\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;t\_6 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\left(t\_2 \cdot t\_7\right) \cdot \left(1 - \left(t\_4 \cdot \left(\left(t\_1 \cdot t\_1\right) \cdot 0.25\right)\right) \cdot \left(t\_4 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\frac{t\_5 \cdot 0.25}{d} \cdot h\right) \cdot t\_5}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* l h)))))
        (t_1 (fmin (fabs M) D))
        (t_2 (sqrt (/ d l)))
        (t_3 (fmax (fabs M) D))
        (t_4 (/ t_3 d))
        (t_5 (* t_1 t_3))
        (t_6
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ t_5 (* 2.0 d)) 2.0)) (/ h l)))))
        (t_7 (sqrt (/ d h))))
   (if (<= t_6 -1e+222)
     (* (* (- 1.0 (/ (* t_5 (* (* (/ t_5 d) 0.25) h)) (* (+ d d) l))) t_2) t_7)
     (if (<= t_6 0.0)
       (*
        t_0
        (-
         1.0
         (* (* (/ t_3 (+ d d)) t_1) (* (* 0.25 (/ (* t_3 t_1) d)) (/ h l)))))
       (if (<= t_6 4e+233)
         (*
          (* t_2 t_7)
          (- 1.0 (* (* t_4 (* (* t_1 t_1) 0.25)) (* t_4 (* 0.5 (/ h l))))))
         (*
          t_0
          (- 1.0 (/ (* (* (/ (* t_5 0.25) d) h) t_5) (* l (+ d d))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((l * h))));
	double t_1 = fmin(fabs(M), D);
	double t_2 = sqrt((d / l));
	double t_3 = fmax(fabs(M), D);
	double t_4 = t_3 / d;
	double t_5 = t_1 * t_3;
	double t_6 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_5 / (2.0 * d)), 2.0)) * (h / l)));
	double t_7 = sqrt((d / h));
	double tmp;
	if (t_6 <= -1e+222) {
		tmp = ((1.0 - ((t_5 * (((t_5 / d) * 0.25) * h)) / ((d + d) * l))) * t_2) * t_7;
	} else if (t_6 <= 0.0) {
		tmp = t_0 * (1.0 - (((t_3 / (d + d)) * t_1) * ((0.25 * ((t_3 * t_1) / d)) * (h / l))));
	} else if (t_6 <= 4e+233) {
		tmp = (t_2 * t_7) * (1.0 - ((t_4 * ((t_1 * t_1) * 0.25)) * (t_4 * (0.5 * (h / l)))));
	} else {
		tmp = t_0 * (1.0 - (((((t_5 * 0.25) / d) * h) * t_5) / (l * (d + d))));
	}
	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) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_0 = abs((d / sqrt((l * h))))
    t_1 = fmin(abs(m), d_1)
    t_2 = sqrt((d / l))
    t_3 = fmax(abs(m), d_1)
    t_4 = t_3 / d
    t_5 = t_1 * t_3
    t_6 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_5 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_7 = sqrt((d / h))
    if (t_6 <= (-1d+222)) then
        tmp = ((1.0d0 - ((t_5 * (((t_5 / d) * 0.25d0) * h)) / ((d + d) * l))) * t_2) * t_7
    else if (t_6 <= 0.0d0) then
        tmp = t_0 * (1.0d0 - (((t_3 / (d + d)) * t_1) * ((0.25d0 * ((t_3 * t_1) / d)) * (h / l))))
    else if (t_6 <= 4d+233) then
        tmp = (t_2 * t_7) * (1.0d0 - ((t_4 * ((t_1 * t_1) * 0.25d0)) * (t_4 * (0.5d0 * (h / l)))))
    else
        tmp = t_0 * (1.0d0 - (((((t_5 * 0.25d0) / d) * h) * t_5) / (l * (d + d))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((l * h))));
	double t_1 = fmin(Math.abs(M), D);
	double t_2 = Math.sqrt((d / l));
	double t_3 = fmax(Math.abs(M), D);
	double t_4 = t_3 / d;
	double t_5 = t_1 * t_3;
	double t_6 = (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_5 / (2.0 * d)), 2.0)) * (h / l)));
	double t_7 = Math.sqrt((d / h));
	double tmp;
	if (t_6 <= -1e+222) {
		tmp = ((1.0 - ((t_5 * (((t_5 / d) * 0.25) * h)) / ((d + d) * l))) * t_2) * t_7;
	} else if (t_6 <= 0.0) {
		tmp = t_0 * (1.0 - (((t_3 / (d + d)) * t_1) * ((0.25 * ((t_3 * t_1) / d)) * (h / l))));
	} else if (t_6 <= 4e+233) {
		tmp = (t_2 * t_7) * (1.0 - ((t_4 * ((t_1 * t_1) * 0.25)) * (t_4 * (0.5 * (h / l)))));
	} else {
		tmp = t_0 * (1.0 - (((((t_5 * 0.25) / d) * h) * t_5) / (l * (d + d))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((l * h))))
	t_1 = fmin(math.fabs(M), D)
	t_2 = math.sqrt((d / l))
	t_3 = fmax(math.fabs(M), D)
	t_4 = t_3 / d
	t_5 = t_1 * t_3
	t_6 = (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_5 / (2.0 * d)), 2.0)) * (h / l)))
	t_7 = math.sqrt((d / h))
	tmp = 0
	if t_6 <= -1e+222:
		tmp = ((1.0 - ((t_5 * (((t_5 / d) * 0.25) * h)) / ((d + d) * l))) * t_2) * t_7
	elif t_6 <= 0.0:
		tmp = t_0 * (1.0 - (((t_3 / (d + d)) * t_1) * ((0.25 * ((t_3 * t_1) / d)) * (h / l))))
	elif t_6 <= 4e+233:
		tmp = (t_2 * t_7) * (1.0 - ((t_4 * ((t_1 * t_1) * 0.25)) * (t_4 * (0.5 * (h / l)))))
	else:
		tmp = t_0 * (1.0 - (((((t_5 * 0.25) / d) * h) * t_5) / (l * (d + d))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(l * h))))
	t_1 = fmin(abs(M), D)
	t_2 = sqrt(Float64(d / l))
	t_3 = fmax(abs(M), D)
	t_4 = Float64(t_3 / d)
	t_5 = Float64(t_1 * t_3)
	t_6 = 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_5 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_7 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_6 <= -1e+222)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(t_5 * Float64(Float64(Float64(t_5 / d) * 0.25) * h)) / Float64(Float64(d + d) * l))) * t_2) * t_7);
	elseif (t_6 <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(t_3 / Float64(d + d)) * t_1) * Float64(Float64(0.25 * Float64(Float64(t_3 * t_1) / d)) * Float64(h / l)))));
	elseif (t_6 <= 4e+233)
		tmp = Float64(Float64(t_2 * t_7) * Float64(1.0 - Float64(Float64(t_4 * Float64(Float64(t_1 * t_1) * 0.25)) * Float64(t_4 * Float64(0.5 * Float64(h / l))))));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(t_5 * 0.25) / d) * h) * t_5) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((l * h))));
	t_1 = min(abs(M), D);
	t_2 = sqrt((d / l));
	t_3 = max(abs(M), D);
	t_4 = t_3 / d;
	t_5 = t_1 * t_3;
	t_6 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((t_5 / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_7 = sqrt((d / h));
	tmp = 0.0;
	if (t_6 <= -1e+222)
		tmp = ((1.0 - ((t_5 * (((t_5 / d) * 0.25) * h)) / ((d + d) * l))) * t_2) * t_7;
	elseif (t_6 <= 0.0)
		tmp = t_0 * (1.0 - (((t_3 / (d + d)) * t_1) * ((0.25 * ((t_3 * t_1) / d)) * (h / l))));
	elseif (t_6 <= 4e+233)
		tmp = (t_2 * t_7) * (1.0 - ((t_4 * ((t_1 * t_1) * 0.25)) * (t_4 * (0.5 * (h / l)))));
	else
		tmp = t_0 * (1.0 - (((((t_5 * 0.25) / d) * h) * t_5) / (l * (d + d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / d), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * t$95$3), $MachinePrecision]}, Block[{t$95$6 = 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$5 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, -1e+222], N[(N[(N[(1.0 - N[(N[(t$95$5 * N[(N[(N[(t$95$5 / d), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$7), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(t$95$0 * N[(1.0 - N[(N[(N[(t$95$3 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(0.25 * N[(N[(t$95$3 * t$95$1), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 4e+233], N[(N[(t$95$2 * t$95$7), $MachinePrecision] * N[(1.0 - N[(N[(t$95$4 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(N[(t$95$5 * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * t$95$5), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)))) < -1e222

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

   5. Applied rewrites 66.5%

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

   if -1e222 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 70.7%

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

   8. Applied rewrites 70.7%

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

   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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.1%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 60.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 60.1%

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

   5. Applied rewrites 60.1%

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

   if 3.99999999999999989e233 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 75.6%

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

   9. Applied rewrites 75.6%

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

Alternative 5: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \left(\left(1 - \frac{t\_3 \cdot \left(\left(\frac{t\_3}{d} \cdot 0.25\right) \cdot h\right)}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ t_5 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{+222}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(\frac{t\_2}{d + d} \cdot t\_1\right) \cdot \left(\left(0.25 \cdot \frac{t\_2 \cdot t\_1}{d}\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\frac{t\_3 \cdot 0.25}{d} \cdot h\right) \cdot t\_3}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* l h)))))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3 (* t_1 t_2))
        (t_4
         (*
          (*
           (- 1.0 (/ (* t_3 (* (* (/ t_3 d) 0.25) h)) (* (+ d d) l)))
           (sqrt (/ d l)))
          (sqrt (/ d h))))
        (t_5
         (*
          (* (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_5 -1e+222)
     t_4
     (if (<= t_5 0.0)
       (*
        t_0
        (-
         1.0
         (* (* (/ t_2 (+ d d)) t_1) (* (* 0.25 (/ (* t_2 t_1) d)) (/ h l)))))
       (if (<= t_5 4e+233)
         t_4
         (*
          t_0
          (- 1.0 (/ (* (* (/ (* t_3 0.25) d) h) t_3) (* l (+ d d))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((l * h))));
	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 = ((1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / ((d + d) * l))) * sqrt((d / l))) * sqrt((d / h));
	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_3 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_5 <= -1e+222) {
		tmp = t_4;
	} else if (t_5 <= 0.0) {
		tmp = t_0 * (1.0 - (((t_2 / (d + d)) * t_1) * ((0.25 * ((t_2 * t_1) / d)) * (h / l))));
	} else if (t_5 <= 4e+233) {
		tmp = t_4;
	} else {
		tmp = t_0 * (1.0 - (((((t_3 * 0.25) / d) * h) * t_3) / (l * (d + d))));
	}
	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) :: t_5
    real(8) :: tmp
    t_0 = abs((d / sqrt((l * h))))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = fmax(abs(m), abs(d_1))
    t_3 = t_1 * t_2
    t_4 = ((1.0d0 - ((t_3 * (((t_3 / d) * 0.25d0) * h)) / ((d + d) * l))) * sqrt((d / l))) * sqrt((d / h))
    t_5 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_3 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_5 <= (-1d+222)) then
        tmp = t_4
    else if (t_5 <= 0.0d0) then
        tmp = t_0 * (1.0d0 - (((t_2 / (d + d)) * t_1) * ((0.25d0 * ((t_2 * t_1) / d)) * (h / l))))
    else if (t_5 <= 4d+233) then
        tmp = t_4
    else
        tmp = t_0 * (1.0d0 - (((((t_3 * 0.25d0) / d) * h) * t_3) / (l * (d + d))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((l * h))));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = fmax(Math.abs(M), Math.abs(D));
	double t_3 = t_1 * t_2;
	double t_4 = ((1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / ((d + d) * l))) * Math.sqrt((d / l))) * Math.sqrt((d / h));
	double t_5 = (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_3 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_5 <= -1e+222) {
		tmp = t_4;
	} else if (t_5 <= 0.0) {
		tmp = t_0 * (1.0 - (((t_2 / (d + d)) * t_1) * ((0.25 * ((t_2 * t_1) / d)) * (h / l))));
	} else if (t_5 <= 4e+233) {
		tmp = t_4;
	} else {
		tmp = t_0 * (1.0 - (((((t_3 * 0.25) / d) * h) * t_3) / (l * (d + d))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((l * h))))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = fmax(math.fabs(M), math.fabs(D))
	t_3 = t_1 * t_2
	t_4 = ((1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / ((d + d) * l))) * math.sqrt((d / l))) * math.sqrt((d / h))
	t_5 = (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_3 / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_5 <= -1e+222:
		tmp = t_4
	elif t_5 <= 0.0:
		tmp = t_0 * (1.0 - (((t_2 / (d + d)) * t_1) * ((0.25 * ((t_2 * t_1) / d)) * (h / l))))
	elif t_5 <= 4e+233:
		tmp = t_4
	else:
		tmp = t_0 * (1.0 - (((((t_3 * 0.25) / d) * h) * t_3) / (l * (d + d))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(l * h))))
	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(1.0 - Float64(Float64(t_3 * Float64(Float64(Float64(t_3 / d) * 0.25) * h)) / Float64(Float64(d + d) * l))) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)))
	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_3 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_5 <= -1e+222)
		tmp = t_4;
	elseif (t_5 <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(t_2 / Float64(d + d)) * t_1) * Float64(Float64(0.25 * Float64(Float64(t_2 * t_1) / d)) * Float64(h / l)))));
	elseif (t_5 <= 4e+233)
		tmp = t_4;
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(t_3 * 0.25) / d) * h) * t_3) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((l * h))));
	t_1 = min(abs(M), abs(D));
	t_2 = max(abs(M), abs(D));
	t_3 = t_1 * t_2;
	t_4 = ((1.0 - ((t_3 * (((t_3 / d) * 0.25) * h)) / ((d + d) * l))) * sqrt((d / l))) * sqrt((d / h));
	t_5 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((t_3 / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_5 <= -1e+222)
		tmp = t_4;
	elseif (t_5 <= 0.0)
		tmp = t_0 * (1.0 - (((t_2 / (d + d)) * t_1) * ((0.25 * ((t_2 * t_1) / d)) * (h / l))));
	elseif (t_5 <= 4e+233)
		tmp = t_4;
	else
		tmp = t_0 * (1.0 - (((((t_3 * 0.25) / d) * h) * t_3) / (l * (d + d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $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[(1.0 - N[(N[(t$95$3 * N[(N[(N[(t$95$3 / d), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $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$3 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1e+222], t$95$4, If[LessEqual[t$95$5, 0.0], N[(t$95$0 * N[(1.0 - N[(N[(N[(t$95$2 / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(0.25 * N[(N[(t$95$2 * t$95$1), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 4e+233], t$95$4, N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(N[(t$95$3 * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * t$95$3), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $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)))) < -1e222 or -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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

   5. Applied rewrites 66.5%

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

   if -1e222 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 70.7%

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

   8. Applied rewrites 70.7%

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

   if 3.99999999999999989e233 < (*.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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-down N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-undiv N/A

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

      16. lift-*.f64 N/A

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

      17. rem-sqrt-square N/A

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

      18. sqrt-fabs-rev N/A

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

      19. lift-sqrt.f64 N/A

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

      20. div-fabs N/A

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

      21. lower-fabs.f64 N/A

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

      22. lower-/.f64 71.3%

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 71.3%

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

   5. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 75.6%

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

   9. Applied rewrites 75.6%

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

Alternative 6: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{d} \cdot D\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2 \cdot \left(1 - \left(t\_0 \cdot M\right) \cdot \left(\left(t\_0 \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(\frac{\left(M \cdot D\right) \cdot 0.25}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 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 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_1 0.0)
     (* t_2 (- 1.0 (* (* t_0 M) (* (* t_0 (* M 0.5)) (/ h l)))))
     (if (<= t_1 5e+144)
       (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h)
       (*
        t_2
        (- 1.0 (/ (* (* (/ (* (* M D) 0.25) d) h) (* M D)) (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 / 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 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2 * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	} else if (t_1 <= 5e+144) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} else {
		tmp = t_2 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))));
	}
	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 = (0.5d0 / d) * d_1
    t_1 = (((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_2 = abs((d / sqrt((l * h))))
    if (t_1 <= 0.0d0) then
        tmp = t_2 * (1.0d0 - ((t_0 * m) * ((t_0 * (m * 0.5d0)) * (h / l))))
    else if (t_1 <= 5d+144) then
        tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h
    else
        tmp = t_2 * (1.0d0 - ((((((m * d_1) * 0.25d0) / d) * h) * (m * d_1)) / (l * (d + d))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 / d) * D;
	double t_1 = (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_2 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2 * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	} else if (t_1 <= 5e+144) {
		tmp = (h * (Math.sqrt((d / h)) * Math.sqrt((d / l)))) / h;
	} else {
		tmp = t_2 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (0.5 / d) * D
	t_1 = (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_2 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2 * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))))
	elif t_1 <= 5e+144:
		tmp = (h * (math.sqrt((d / h)) * math.sqrt((d / l)))) / h
	else:
		tmp = t_2 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 / 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 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(t_0 * M) * Float64(Float64(t_0 * Float64(M * 0.5)) * Float64(h / l)))));
	elseif (t_1 <= 5e+144)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	else
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) * 0.25) / d) * h) * Float64(M * D)) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 / d) * D;
	t_1 = (((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_2 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_2 * (1.0 - ((t_0 * M) * ((t_0 * (M * 0.5)) * (h / l))));
	elseif (t_1 <= 5e+144)
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	else
		tmp = t_2 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * D), $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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$2 * N[(1.0 - N[(N[(t$95$0 * M), $MachinePrecision] * N[(N[(t$95$0 * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+144], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$2 * N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $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)))) < -0.0

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\frac{D}{d + d}} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. mult-flip N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(\frac{1}{d + d} \cdot D\right)} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(\frac{1}{d + d} \cdot D\right)} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{1}{\color{blue}{d + d}} \cdot D\right) \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. count-2-rev N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot D\right) \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot D\right) \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lower-/.f64 71.3%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\color{blue}{\frac{0.5}{d}} \cdot D\right) \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   7. Applied rewrites 71.3%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\left(\frac{0.5}{d} \cdot D\right)} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\color{blue}{\frac{D}{d + d}} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. mult-flip N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\color{blue}{\left(D \cdot \frac{1}{d + d}\right)} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{d + d} \cdot D\right)} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{d + d} \cdot D\right)} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\left(\frac{1}{\color{blue}{d + d}} \cdot D\right) \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. count-2-rev N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot D\right) \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot D\right) \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\left(\frac{\color{blue}{\frac{1}{2}}}{d} \cdot D\right) \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lower-/.f64 71.4%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\left(\color{blue}{\frac{0.5}{d}} \cdot D\right) \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   9. Applied rewrites 71.4%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(\left(\color{blue}{\left(\frac{0.5}{d} \cdot D\right)} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   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)))) < 4.9999999999999999e144

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   if 4.9999999999999999e144 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right)} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\left(\color{blue}{\frac{D}{d}} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\color{blue}{\frac{D \cdot M}{d}} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{1}{4}}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{1}{4}}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. lower-*.f64 75.6%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot 0.25}}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 75.6%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot 0.25}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot M\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(t\_0 \cdot 0.25\right), t\_0 \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(\frac{\left(M \cdot D\right) \cdot 0.25}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M))
        (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 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_1 0.0)
     (* t_2 (fma (* (/ h l) (* t_0 0.25)) (* t_0 -0.5) 1.0))
     (if (<= t_1 5e+144)
       (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h)
       (*
        t_2
        (- 1.0 (/ (* (* (/ (* (* M D) 0.25) d) h) (* M D)) (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * M;
	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 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2 * fma(((h / l) * (t_0 * 0.25)), (t_0 * -0.5), 1.0);
	} else if (t_1 <= 5e+144) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} else {
		tmp = t_2 * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * M)
	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 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_2 * fma(Float64(Float64(h / l) * Float64(t_0 * 0.25)), Float64(t_0 * -0.5), 1.0));
	elseif (t_1 <= 5e+144)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	else
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) * 0.25) / d) * h) * Float64(M * D)) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M), $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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$2 * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+144], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$2 * N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $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)))) < -0.0

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{D}{d + d} \cdot M\right)\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{D}{d + d} \cdot M\right)\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(\color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\mathsf{neg}\left(\frac{D}{d + d} \cdot M\right)\right)} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}, \mathsf{neg}\left(\frac{D}{d + d} \cdot M\right), 1\right)} \]

   7. Applied rewrites 71.4%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right), \left(\frac{D}{d} \cdot M\right) \cdot -0.5, 1\right)} \]

   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)))) < 4.9999999999999999e144

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   if 4.9999999999999999e144 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right)} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\left(\color{blue}{\frac{D}{d}} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\color{blue}{\frac{D \cdot M}{d}} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{1}{4}}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{1}{4}}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. lower-*.f64 75.6%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot 0.25}}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 75.6%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot 0.25}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{\left(M \cdot D\right) \cdot 0.25}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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
         (*
          (fabs (/ d (sqrt (* l h))))
          (- 1.0 (/ (* (* (/ (* (* M D) 0.25) d) h) (* M D)) (* l (+ d d)))))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 5e+144) (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h) 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 = fabs((d / sqrt((l * h)))) * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+144) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} 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 = abs((d / sqrt((l * h)))) * (1.0d0 - ((((((m * d_1) * 0.25d0) / d) * h) * (m * d_1)) / (l * (d + d))))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+144) then
        tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h
    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.abs((d / Math.sqrt((l * h)))) * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+144) {
		tmp = (h * (Math.sqrt((d / h)) * Math.sqrt((d / l)))) / h;
	} 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.fabs((d / math.sqrt((l * h)))) * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+144:
		tmp = (h * (math.sqrt((d / h)) * math.sqrt((d / l)))) / h
	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(abs(Float64(d / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * D) * 0.25) / d) * h) * Float64(M * D)) / Float64(l * Float64(d + d)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+144)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	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 = abs((d / sqrt((l * h)))) * (1.0 - ((((((M * D) * 0.25) / d) * h) * (M * D)) / (l * (d + d))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+144)
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+144], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 4.9999999999999999e144 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right)} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\left(\color{blue}{\frac{D}{d}} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\color{blue}{\frac{D \cdot M}{d}} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{1}{4}}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{1}{4}}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. lower-*.f64 75.6%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot 0.25}}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 75.6%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot 0.25}{d}} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   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)))) < 4.9999999999999999e144

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \ell \cdot \left(d + d\right)\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{t\_1}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{t\_1}\right)\\ \end{array} \]

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 (* l (+ d d)))
        (t_2 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_0 0.0)
     (* t_2 (- 1.0 (/ (* (* (* (* (/ D d) M) 0.25) h) (* M D)) t_1)))
     (if (<= t_0 4e+233)
       (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h)
       (* t_2 (- 1.0 (/ (* (* 0.25 (/ (* D (* M h)) d)) (* M D)) 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 = l * (d + d);
	double t_2 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2 * (1.0 - ((((((D / d) * M) * 0.25) * h) * (M * D)) / t_1));
	} else if (t_0 <= 4e+233) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} else {
		tmp = t_2 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / 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) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = l * (d + d)
    t_2 = abs((d / sqrt((l * h))))
    if (t_0 <= 0.0d0) then
        tmp = t_2 * (1.0d0 - ((((((d_1 / d) * m) * 0.25d0) * h) * (m * d_1)) / t_1))
    else if (t_0 <= 4d+233) then
        tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h
    else
        tmp = t_2 * (1.0d0 - (((0.25d0 * ((d_1 * (m * h)) / d)) * (m * d_1)) / 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 = l * (d + d);
	double t_2 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2 * (1.0 - ((((((D / d) * M) * 0.25) * h) * (M * D)) / t_1));
	} else if (t_0 <= 4e+233) {
		tmp = (h * (Math.sqrt((d / h)) * Math.sqrt((d / l)))) / h;
	} else {
		tmp = t_2 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / 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 = l * (d + d)
	t_2 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_2 * (1.0 - ((((((D / d) * M) * 0.25) * h) * (M * D)) / t_1))
	elif t_0 <= 4e+233:
		tmp = (h * (math.sqrt((d / h)) * math.sqrt((d / l)))) / h
	else:
		tmp = t_2 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / 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(l * Float64(d + d))
	t_2 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D / d) * M) * 0.25) * h) * Float64(M * D)) / t_1)));
	elseif (t_0 <= 4e+233)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	else
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(Float64(D * Float64(M * h)) / d)) * Float64(M * D)) / 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 = l * (d + d);
	t_2 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_2 * (1.0 - ((((((D / d) * M) * 0.25) * h) * (M * D)) / t_1));
	elseif (t_0 <= 4e+233)
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	else
		tmp = t_2 * (1.0 - (((0.25 * ((D * (M * h)) / d)) * (M * D)) / 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[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(t$95$2 * N[(1.0 - N[(N[(N[(N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+233], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$2 * N[(1.0 - N[(N[(N[(0.25 * N[(N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]

   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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   if 3.99999999999999989e233 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]

   8. Taylor expanded in d around 0

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)} \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{1}{4} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d}}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d}}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lower-*.f64 73.4%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   10. Applied rewrites 73.4%

       \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)} \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := D \cdot \left(M \cdot h\right)\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_2 \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(0.25 \cdot \frac{t\_1}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(0.25 \cdot \frac{t\_1}{d}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

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 (* D (* M h)))
        (t_2 (fabs (/ d (sqrt (* l h))))))
   (if (<= t_0 0.0)
     (* t_2 (- 1.0 (* (* (/ D (+ d d)) M) (* 0.25 (/ t_1 (* d l))))))
     (if (<= t_0 4e+233)
       (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h)
       (* t_2 (- 1.0 (/ (* (* 0.25 (/ t_1 d)) (* M D)) (* l (+ d d)))))))))

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 = D * (M * h);
	double t_2 = fabs((d / sqrt((l * h))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2 * (1.0 - (((D / (d + d)) * M) * (0.25 * (t_1 / (d * l)))));
	} else if (t_0 <= 4e+233) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} else {
		tmp = t_2 * (1.0 - (((0.25 * (t_1 / d)) * (M * D)) / (l * (d + d))));
	}
	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 = (((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 = d_1 * (m * h)
    t_2 = abs((d / sqrt((l * h))))
    if (t_0 <= 0.0d0) then
        tmp = t_2 * (1.0d0 - (((d_1 / (d + d)) * m) * (0.25d0 * (t_1 / (d * l)))))
    else if (t_0 <= 4d+233) then
        tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h
    else
        tmp = t_2 * (1.0d0 - (((0.25d0 * (t_1 / d)) * (m * d_1)) / (l * (d + d))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (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 = D * (M * h);
	double t_2 = Math.abs((d / Math.sqrt((l * h))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2 * (1.0 - (((D / (d + d)) * M) * (0.25 * (t_1 / (d * l)))));
	} else if (t_0 <= 4e+233) {
		tmp = (h * (Math.sqrt((d / h)) * Math.sqrt((d / l)))) / h;
	} else {
		tmp = t_2 * (1.0 - (((0.25 * (t_1 / d)) * (M * D)) / (l * (d + d))));
	}
	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 = D * (M * h)
	t_2 = math.fabs((d / math.sqrt((l * h))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_2 * (1.0 - (((D / (d + d)) * M) * (0.25 * (t_1 / (d * l)))))
	elif t_0 <= 4e+233:
		tmp = (h * (math.sqrt((d / h)) * math.sqrt((d / l)))) / h
	else:
		tmp = t_2 * (1.0 - (((0.25 * (t_1 / d)) * (M * D)) / (l * (d + d))))
	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(D * Float64(M * h))
	t_2 = abs(Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(D / Float64(d + d)) * M) * Float64(0.25 * Float64(t_1 / Float64(d * l))))));
	elseif (t_0 <= 4e+233)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	else
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(t_1 / d)) * Float64(M * D)) / Float64(l * Float64(d + d)))));
	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 = D * (M * h);
	t_2 = abs((d / sqrt((l * h))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_2 * (1.0 - (((D / (d + d)) * M) * (0.25 * (t_1 / (d * l)))));
	elseif (t_0 <= 4e+233)
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	else
		tmp = t_2 * (1.0 - (((0.25 * (t_1 / d)) * (M * D)) / (l * (d + d))));
	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[(D * N[(M * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(t$95$2 * N[(1.0 - N[(N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(0.25 * N[(t$95$1 / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+233], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$2 * N[(1.0 - N[(N[(N[(0.25 * N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $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)))) < -0.0

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d} \cdot \ell}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)\right) \]

      5. lower-*.f64 73.2%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\ell}}\right)\right) \]

   8. Applied rewrites 73.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]

   if -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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   if 3.99999999999999989e233 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]

   8. Taylor expanded in d around 0

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)} \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{1}{4} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d}}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d}}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lower-*.f64 73.4%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   10. Applied rewrites 73.4%

       \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)} \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{\mathsf{max}\left(M, D\right)}{d + d} \cdot \mathsf{min}\left(M, D\right)\right) \cdot \left(0.25 \cdot \frac{\mathsf{max}\left(M, D\right) \cdot \left(\mathsf{min}\left(M, D\right) \cdot h\right)}{d \cdot \ell}\right)\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (fabs (/ d (sqrt (* l h))))
          (-
           1.0
           (*
            (* (/ (fmax M D) (+ d d)) (fmin M D))
            (* 0.25 (/ (* (fmax M D) (* (fmin M D) h)) (* d l)))))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 4e+233) (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((l * h)))) * (1.0 - (((fmax(M, D) / (d + d)) * fmin(M, D)) * (0.25 * ((fmax(M, D) * (fmin(M, D) * h)) / (d * l)))));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 4e+233) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} else {
		tmp = t_0;
	}
	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 = abs((d / sqrt((l * h)))) * (1.0d0 - (((fmax(m, d_1) / (d + d)) * fmin(m, d_1)) * (0.25d0 * ((fmax(m, d_1) * (fmin(m, d_1) * h)) / (d * l)))))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((fmin(m, d_1) * fmax(m, d_1)) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 4d+233) then
        tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((l * h)))) * (1.0 - (((fmax(M, D) / (d + d)) * fmin(M, D)) * (0.25 * ((fmax(M, D) * (fmin(M, D) * h)) / (d * l)))));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 4e+233) {
		tmp = (h * (Math.sqrt((d / h)) * Math.sqrt((d / l)))) / h;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((l * h)))) * (1.0 - (((fmax(M, D) / (d + d)) * fmin(M, D)) * (0.25 * ((fmax(M, D) * (fmin(M, D) * h)) / (d * l)))))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 4e+233:
		tmp = (h * (math.sqrt((d / h)) * math.sqrt((d / l)))) / h
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(fmax(M, D) / Float64(d + d)) * fmin(M, D)) * Float64(0.25 * Float64(Float64(fmax(M, D) * Float64(fmin(M, D) * h)) / Float64(d * l))))))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 4e+233)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((l * h)))) * (1.0 - (((max(M, D) / (d + d)) * min(M, D)) * (0.25 * ((max(M, D) * (min(M, D) * h)) / (d * l)))));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 4e+233)
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Max[M, D], $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(N[Max[M, D], $MachinePrecision] * N[(N[Min[M, D], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 4e+233], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 3.99999999999999989e233 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d} \cdot \ell}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)\right) \]

      5. lower-*.f64 73.2%

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\ell}}\right)\right) \]

   8. Applied rewrites 73.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]

   if -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)))) < 3.99999999999999989e233

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{fma}\left(\left(\left(\frac{t\_1}{d} \cdot t\_0\right) \cdot t\_0\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot t\_1}{\ell \cdot d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fmin (fabs M) (fabs D)))
        (t_1 (fmax (fabs M) (fabs D)))
        (t_2
         (*
          (fma
           (* (* (* (/ t_1 d) t_0) t_0) -0.25)
           (/ (* (* 0.5 h) t_1) (* l d))
           1.0)
          (/ (fabs d) (sqrt (* l h)))))
        (t_3
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_3 0.0)
     t_2
     (if (<= t_3 4e+233) (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double t_2 = fma(((((t_1 / d) * t_0) * t_0) * -0.25), (((0.5 * h) * t_1) / (l * d)), 1.0) * (fabs(d) / sqrt((l * h)));
	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_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 4e+233) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	t_2 = Float64(fma(Float64(Float64(Float64(Float64(t_1 / d) * t_0) * t_0) * -0.25), Float64(Float64(Float64(0.5 * h) * t_1) / Float64(l * d)), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	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_0 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 4e+233)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(t$95$1 / d), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(N[(0.5 * h), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $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$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 4e+233], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 3.99999999999999989e233 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{\sqrt{\ell \cdot h}} \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{\sqrt{\ell \cdot h}} \]

      3. rem-sqrt-square N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{\color{blue}{\left|-d\right|}}{\sqrt{\ell \cdot h}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\sqrt{\ell \cdot h}} \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

      6. lower-fabs.f64 69.9%

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 69.9%

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 3.99999999999999989e233

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \frac{t\_2}{d}\\ t_4 := t\_1 \cdot t\_2\\ \mathbf{if}\;h \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t\_2 \cdot \left(\left(\left(0.5 \cdot h\right) \cdot t\_1\right) \cdot \left(t\_1 \cdot t\_3\right)\right)}{\ell \cdot d}, -0.25, 1\right) \cdot d}{-t\_0}\\ \mathbf{elif}\;h \leq 4.2 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_4 \cdot \left(-0.25 \cdot t\_1\right)}{d}, \frac{\left(0.5 \cdot h\right) \cdot t\_2}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - \frac{\left(\left(\left(t\_3 \cdot t\_1\right) \cdot 0.25\right) \cdot h\right) \cdot t\_4}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l)))
        (t_1 (fmin (fabs M) (fabs D)))
        (t_2 (fmax (fabs M) (fabs D)))
        (t_3 (/ t_2 d))
        (t_4 (* t_1 t_2)))
   (if (<= h -5e+30)
     (/
      (*
       (fma (/ (* t_2 (* (* (* 0.5 h) t_1) (* t_1 t_3))) (* l d)) -0.25 1.0)
       d)
      (- t_0))
     (if (<= h 4.2e-308)
       (*
        (fma (/ (* t_4 (* -0.25 t_1)) d) (/ (* (* 0.5 h) t_2) (* l d)) 1.0)
        (/ (- d) (sqrt (* l h))))
       (*
        (/ d t_0)
        (- 1.0 (/ (* (* (* (* t_3 t_1) 0.25) h) t_4) (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = t_2 / d;
	double t_4 = t_1 * t_2;
	double tmp;
	if (h <= -5e+30) {
		tmp = (fma(((t_2 * (((0.5 * h) * t_1) * (t_1 * t_3))) / (l * d)), -0.25, 1.0) * d) / -t_0;
	} else if (h <= 4.2e-308) {
		tmp = fma(((t_4 * (-0.25 * t_1)) / d), (((0.5 * h) * t_2) / (l * d)), 1.0) * (-d / sqrt((l * h)));
	} else {
		tmp = (d / t_0) * (1.0 - (((((t_3 * t_1) * 0.25) * h) * t_4) / (l * (d + d))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(t_2 / d)
	t_4 = Float64(t_1 * t_2)
	tmp = 0.0
	if (h <= -5e+30)
		tmp = Float64(Float64(fma(Float64(Float64(t_2 * Float64(Float64(Float64(0.5 * h) * t_1) * Float64(t_1 * t_3))) / Float64(l * d)), -0.25, 1.0) * d) / Float64(-t_0));
	elseif (h <= 4.2e-308)
		tmp = Float64(fma(Float64(Float64(t_4 * Float64(-0.25 * t_1)) / d), Float64(Float64(Float64(0.5 * h) * t_2) / Float64(l * d)), 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(t_3 * t_1) * 0.25) * h) * t_4) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $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$2 / d), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$2), $MachinePrecision]}, If[LessEqual[h, -5e+30], N[(N[(N[(N[(N[(t$95$2 * N[(N[(N[(0.5 * h), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * d), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[h, 4.2e-308], N[(N[(N[(N[(t$95$4 * N[(-0.25 * t$95$1), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(0.5 * h), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(t$95$3 * t$95$1), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision] * t$95$4), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.9999999999999998e30

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   7. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), -0.25, 1\right) \cdot d}{-\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{D}{\ell \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{D}{\ell \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D}{\ell \cdot d} \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D}{\ell \cdot d}} \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right), \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)}{\ell \cdot d}}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)}{\ell \cdot d}}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

   9. Applied rewrites 38.1%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D \cdot \left(\left(\left(0.5 \cdot h\right) \cdot M\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}{\ell \cdot d}}, -0.25, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

   if -4.9999999999999998e30 < h < 4.2e-308

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)} \cdot \frac{-1}{4}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{D}{d} \cdot M\right) \cdot \left(M \cdot \frac{-1}{4}\right)}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \left(M \cdot \frac{-1}{4}\right), \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{D}{d}} \cdot M\right) \cdot \left(M \cdot \frac{-1}{4}\right), \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{D \cdot M}{d}} \cdot \left(M \cdot \frac{-1}{4}\right), \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \left(M \cdot \frac{-1}{4}\right), \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot \frac{-1}{4}\right)}{d}}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot \frac{-1}{4}\right)}{d}}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot \frac{-1}{4}\right)}}{d}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(M \cdot D\right)} \cdot \left(M \cdot \frac{-1}{4}\right)}{d}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot M\right)}}{d}, \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      13. lower-*.f64 36.0%

          \[\leadsto \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \color{blue}{\left(-0.25 \cdot M\right)}}{d}, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 36.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(-0.25 \cdot M\right)}{d}}, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   if 4.2e-308 < h

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. neg-fabs N/A

         \[\leadsto \color{blue}{\left|\mathsf{neg}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{-d}}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{-\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      11. fabs-div N/A

          \[\leadsto \color{blue}{\frac{\left|\mathsf{neg}\left(\left(-d\right)\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\left|\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto \frac{\left|\color{blue}{d}\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      14. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(-d\right)}\right)}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      20. neg-fabs N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      22. sqrt-fabs-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      23. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      24. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)\right)\right)}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{D \cdot \left(\left(\left(0.5 \cdot h\right) \cdot M\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}{\ell \cdot d}, -0.25, 1\right) \cdot d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= l -4e+197)
     (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l -1.2e-306)
       (/
        (*
         (fma (/ (* D (* (* (* 0.5 h) M) (* M (/ D d)))) (* l d)) -0.25 1.0)
         d)
        (- t_0))
       (*
        (/ d t_0)
        (- 1.0 (/ (* (* (* (* (/ D d) M) 0.25) h) (* M D)) (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -4e+197) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -1.2e-306) {
		tmp = (fma(((D * (((0.5 * h) * M) * (M * (D / d)))) / (l * d)), -0.25, 1.0) * d) / -t_0;
	} else {
		tmp = (d / t_0) * (1.0 - ((((((D / d) * M) * 0.25) * h) * (M * D)) / (l * (d + d))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -4e+197)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -1.2e-306)
		tmp = Float64(Float64(fma(Float64(Float64(D * Float64(Float64(Float64(0.5 * h) * M) * Float64(M * Float64(D / d)))) / Float64(l * d)), -0.25, 1.0) * d) / Float64(-t_0));
	else
		tmp = Float64(Float64(d / t_0) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D / d) * M) * 0.25) * h) * Float64(M * D)) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e+197], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.2e-306], N[(N[(N[(N[(N[(D * N[(N[(N[(0.5 * h), $MachinePrecision] * M), $MachinePrecision] * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * d), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.9999999999999998e197

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-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 37.7%

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 37.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(\color{blue}{h}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      8. lower-neg.f64 22.2%

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}} \]

   if -3.9999999999999998e197 < l < -1.2e-306

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   7. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), -0.25, 1\right) \cdot d}{-\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{D}{\ell \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{D}{\ell \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D}{\ell \cdot d} \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D}{\ell \cdot d}} \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right), \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)}{\ell \cdot d}}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D \cdot \left(\left(\frac{1}{2} \cdot h\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right)}{\ell \cdot d}}, \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

   9. Applied rewrites 38.1%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{D \cdot \left(\left(\left(0.5 \cdot h\right) \cdot M\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}{\ell \cdot d}}, -0.25, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

   if -1.2e-306 < l

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. neg-fabs N/A

         \[\leadsto \color{blue}{\left|\mathsf{neg}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{-d}}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{-\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      11. fabs-div N/A

          \[\leadsto \color{blue}{\frac{\left|\mathsf{neg}\left(\left(-d\right)\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\left|\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto \frac{\left|\color{blue}{d}\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      14. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(-d\right)}\right)}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      20. neg-fabs N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      22. sqrt-fabs-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      23. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      24. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)\right)\right)}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot M\\ t_1 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(t\_0 \cdot M\right), -0.25, 1\right) \cdot d}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_1} \cdot \left(1 - \frac{\left(\left(t\_0 \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M)) (t_1 (sqrt (* h l))))
   (if (<= l -4e+197)
     (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l -3e-307)
       (/
        (* (fma (* (* (/ D (* l d)) (* 0.5 h)) (* t_0 M)) -0.25 1.0) d)
        (- t_1))
       (*
        (/ d t_1)
        (- 1.0 (/ (* (* (* t_0 0.25) h) (* M D)) (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * M;
	double t_1 = sqrt((h * l));
	double tmp;
	if (l <= -4e+197) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -3e-307) {
		tmp = (fma((((D / (l * d)) * (0.5 * h)) * (t_0 * M)), -0.25, 1.0) * d) / -t_1;
	} else {
		tmp = (d / t_1) * (1.0 - ((((t_0 * 0.25) * h) * (M * D)) / (l * (d + d))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * M)
	t_1 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -4e+197)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -3e-307)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(D / Float64(l * d)) * Float64(0.5 * h)) * Float64(t_0 * M)), -0.25, 1.0) * d) / Float64(-t_1));
	else
		tmp = Float64(Float64(d / t_1) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * 0.25) * h) * Float64(M * D)) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e+197], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -3e-307], N[(N[(N[(N[(N[(N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * M), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * d), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(d / t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * 0.25), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.9999999999999998e197

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-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 37.7%

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 37.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(\color{blue}{h}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      8. lower-neg.f64 22.2%

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}} \]

   if -3.9999999999999998e197 < l < -2.9999999999999999e-307

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   7. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), -0.25, 1\right) \cdot d}{-\sqrt{h \cdot \ell}}} \]

   if -2.9999999999999999e-307 < l

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. neg-fabs N/A

         \[\leadsto \color{blue}{\left|\mathsf{neg}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{-d}}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{-\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      11. fabs-div N/A

          \[\leadsto \color{blue}{\frac{\left|\mathsf{neg}\left(\left(-d\right)\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\left|\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto \frac{\left|\color{blue}{d}\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      14. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(-d\right)}\right)}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      20. neg-fabs N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      22. sqrt-fabs-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      23. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      24. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)\right)\right)}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot M\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(t\_0 \cdot M\right), -0.25, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\left(t\_0 \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M)))
   (if (<= l -2e+98)
     (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l -3e-307)
       (*
        (fma (* (* (/ D (* l d)) (* 0.5 h)) (* t_0 M)) -0.25 1.0)
        (/ (- d) (sqrt (* l h))))
       (*
        (/ d (sqrt (* h l)))
        (- 1.0 (/ (* (* (* t_0 0.25) h) (* M D)) (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * M;
	double tmp;
	if (l <= -2e+98) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -3e-307) {
		tmp = fma((((D / (l * d)) * (0.5 * h)) * (t_0 * M)), -0.25, 1.0) * (-d / sqrt((l * h)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - ((((t_0 * 0.25) * h) * (M * D)) / (l * (d + d))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * M)
	tmp = 0.0
	if (l <= -2e+98)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -3e-307)
		tmp = Float64(fma(Float64(Float64(Float64(D / Float64(l * d)) * Float64(0.5 * h)) * Float64(t_0 * M)), -0.25, 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * 0.25) * h) * Float64(M * D)) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[l, -2e+98], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -3e-307], N[(N[(N[(N[(N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * M), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * 0.25), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2e98

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-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 37.7%

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 37.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(\color{blue}{h}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      8. lower-neg.f64 22.2%

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}} \]

   if -2e98 < l < -2.9999999999999999e-307

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) \cdot \frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d} + 1\right)} \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d} \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right)} + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\ell \cdot d}} \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(\frac{1}{2} \cdot h\right) \cdot D}}{\ell \cdot d} \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{\left(\frac{1}{2} \cdot h\right) \cdot D}{\color{blue}{\ell \cdot d}} \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      6. times-frac N/A

         \[\leadsto \left(\color{blue}{\left(\frac{\frac{1}{2} \cdot h}{\ell} \cdot \frac{D}{d}\right)} \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\frac{1}{2} \cdot h}}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      8. associate-*r/ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{h}{\ell}\right)} \cdot \frac{D}{d}\right) \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{2} \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \frac{D}{d}\right) \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{h}{\ell}\right)} \cdot \frac{D}{d}\right) \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{1}{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\frac{D}{d}}\right) \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      12. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)} \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)} \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right) + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \cdot \color{blue}{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \frac{-1}{4}\right)} + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right)\right) \cdot \frac{-1}{4}} + 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), \frac{-1}{4}, 1\right)} \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), -0.25, 1\right)} \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   if -2.9999999999999999e-307 < l

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. neg-fabs N/A

         \[\leadsto \color{blue}{\left|\mathsf{neg}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{-d}}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{-\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      11. fabs-div N/A

          \[\leadsto \color{blue}{\frac{\left|\mathsf{neg}\left(\left(-d\right)\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\left|\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto \frac{\left|\color{blue}{d}\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      14. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(-d\right)}\right)}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      20. neg-fabs N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      22. sqrt-fabs-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      23. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      24. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)\right)\right)}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 71.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\\ t_1 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(0.5 \cdot h\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot t\_0}{\left(\ell \cdot d\right) \cdot d} \cdot \mathsf{min}\left(M, D\right), -0.25, 1\right) \cdot d}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_1} \cdot \left(1 - \frac{\left(\left(\left(\frac{\mathsf{max}\left(M, D\right)}{d} \cdot \mathsf{min}\left(M, D\right)\right) \cdot 0.25\right) \cdot h\right) \cdot t\_0}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (fmin M D) (fmax M D))) (t_1 (sqrt (* h l))))
   (if (<= l -4e+197)
     (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l -3e-307)
       (/
        (*
         (fma
          (* (/ (* (* (* 0.5 h) (fmax M D)) t_0) (* (* l d) d)) (fmin M D))
          -0.25
          1.0)
         d)
        (- t_1))
       (*
        (/ d t_1)
        (-
         1.0
         (/
          (* (* (* (* (/ (fmax M D) d) (fmin M D)) 0.25) h) t_0)
          (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, D) * fmax(M, D);
	double t_1 = sqrt((h * l));
	double tmp;
	if (l <= -4e+197) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -3e-307) {
		tmp = (fma((((((0.5 * h) * fmax(M, D)) * t_0) / ((l * d) * d)) * fmin(M, D)), -0.25, 1.0) * d) / -t_1;
	} else {
		tmp = (d / t_1) * (1.0 - ((((((fmax(M, D) / d) * fmin(M, D)) * 0.25) * h) * t_0) / (l * (d + d))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(fmin(M, D) * fmax(M, D))
	t_1 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -4e+197)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -3e-307)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(0.5 * h) * fmax(M, D)) * t_0) / Float64(Float64(l * d) * d)) * fmin(M, D)), -0.25, 1.0) * d) / Float64(-t_1));
	else
		tmp = Float64(Float64(d / t_1) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(fmax(M, D) / d) * fmin(M, D)) * 0.25) * h) * t_0) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e+197], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -3e-307], N[(N[(N[(N[(N[(N[(N[(N[(0.5 * h), $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * d), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(d / t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.9999999999999998e197

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-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 37.7%

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 37.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(\color{blue}{h}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      8. lower-neg.f64 22.2%

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}} \]

   if -3.9999999999999998e197 < l < -2.9999999999999999e-307

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   7. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), -0.25, 1\right) \cdot d}{-\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), \frac{-1}{4}, 1\right) \cdot d}{-\sqrt{h \cdot \ell}} \]

   9. Applied rewrites 35.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(0.5 \cdot h\right) \cdot D\right) \cdot \left(M \cdot D\right)}{\left(\ell \cdot d\right) \cdot d} \cdot M, -0.25, 1\right)} \cdot d}{-\sqrt{h \cdot \ell}} \]

   if -2.9999999999999999e-307 < l

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. neg-fabs N/A

         \[\leadsto \color{blue}{\left|\mathsf{neg}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{-d}}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{-\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      11. fabs-div N/A

          \[\leadsto \color{blue}{\frac{\left|\mathsf{neg}\left(\left(-d\right)\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\left|\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto \frac{\left|\color{blue}{d}\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      14. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(-d\right)}\right)}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      20. neg-fabs N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      22. sqrt-fabs-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      23. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      24. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)\right)\right)}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 61.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\frac{t\_0 \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-306}:\\ \;\;\;\;\frac{t\_0 \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\left(\left(\frac{\mathsf{max}\left(M, D\right)}{d} \cdot \mathsf{min}\left(M, D\right)\right) \cdot 0.25\right) \cdot h\right) \cdot \left(\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\right)}{\ell \cdot \left(d + d\right)}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -5e-153)
     (/ (* t_0 (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l -1.25e-306)
       (/ (* t_0 (* -1.0 (* d (sqrt (* -1.0 (/ h (* d l))))))) h)
       (*
        (/ d (sqrt (* h l)))
        (-
         1.0
         (/
          (*
           (* (* (* (/ (fmax M D) d) (fmin M D)) 0.25) h)
           (* (fmin M D) (fmax M D)))
          (* l (+ d d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -5e-153) {
		tmp = (t_0 * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -1.25e-306) {
		tmp = (t_0 * (-1.0 * (d * sqrt((-1.0 * (h / (d * l))))))) / h;
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - ((((((fmax(M, D) / d) * fmin(M, D)) * 0.25) * h) * (fmin(M, D) * fmax(M, D))) / (l * (d + d))));
	}
	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)
    if (l <= (-5d-153)) then
        tmp = (t_0 * sqrt((d / l))) / sqrt(-h)
    else if (l <= (-1.25d-306)) then
        tmp = (t_0 * ((-1.0d0) * (d * sqrt(((-1.0d0) * (h / (d * l))))))) / h
    else
        tmp = (d / sqrt((h * l))) * (1.0d0 - ((((((fmax(m, d_1) / d) * fmin(m, d_1)) * 0.25d0) * h) * (fmin(m, d_1) * fmax(m, d_1))) / (l * (d + d))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -5e-153) {
		tmp = (t_0 * Math.sqrt((d / l))) / Math.sqrt(-h);
	} else if (l <= -1.25e-306) {
		tmp = (t_0 * (-1.0 * (d * Math.sqrt((-1.0 * (h / (d * l))))))) / h;
	} else {
		tmp = (d / Math.sqrt((h * l))) * (1.0 - ((((((fmax(M, D) / d) * fmin(M, D)) * 0.25) * h) * (fmin(M, D) * fmax(M, D))) / (l * (d + d))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -5e-153:
		tmp = (t_0 * math.sqrt((d / l))) / math.sqrt(-h)
	elif l <= -1.25e-306:
		tmp = (t_0 * (-1.0 * (d * math.sqrt((-1.0 * (h / (d * l))))))) / h
	else:
		tmp = (d / math.sqrt((h * l))) * (1.0 - ((((((fmax(M, D) / d) * fmin(M, D)) * 0.25) * h) * (fmin(M, D) * fmax(M, D))) / (l * (d + d))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -5e-153)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -1.25e-306)
		tmp = Float64(Float64(t_0 * Float64(-1.0 * Float64(d * sqrt(Float64(-1.0 * Float64(h / Float64(d * l))))))) / h);
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(fmax(M, D) / d) * fmin(M, D)) * 0.25) * h) * Float64(fmin(M, D) * fmax(M, D))) / Float64(l * Float64(d + d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -5e-153)
		tmp = (t_0 * sqrt((d / l))) / sqrt(-h);
	elseif (l <= -1.25e-306)
		tmp = (t_0 * (-1.0 * (d * sqrt((-1.0 * (h / (d * l))))))) / h;
	else
		tmp = (d / sqrt((h * l))) * (1.0 - ((((((max(M, D) / d) * min(M, D)) * 0.25) * h) * (min(M, D) * max(M, D))) / (l * (d + d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -5e-153], N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.25e-306], N[(N[(t$95$0 * N[(-1.0 * N[(d * N[Sqrt[N[(-1.0 * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] / d), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision] * N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.00000000000000033e-153

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-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 37.7%

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 37.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(\color{blue}{h}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      8. lower-neg.f64 22.2%

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}} \]

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}} \]

   if -5.00000000000000033e-153 < l < -1.25e-306

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{-d}{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. sqrt-undiv N/A

          \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{-d}}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{-h}}} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{-h}}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{-h}}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{\color{blue}{\sqrt{\frac{d}{\ell}}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{-h}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. sqrt-undiv N/A

          \[\leadsto \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{-h}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{-h}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 29.7%

      \[\leadsto \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{\frac{d}{\ell}}{-h}}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      8. lower-*.f64 5.9%

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

   8. Applied rewrites 5.9%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h}} \]

   9. Taylor expanded in d around -inf

      \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]

       6. lower-*.f64 9.3%

          \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]

   11. Applied rewrites 9.3%

       \[\leadsto \frac{\sqrt{-d} \cdot \left(-1 \cdot \left(d \cdot \sqrt{-1 \cdot \frac{h}{d \cdot \ell}}\right)\right)}{h} \]

   if -1.25e-306 < l

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell}} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      10. frac-times N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}}\right) \]

   7. Applied rewrites 74.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{\left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      2. neg-fabs N/A

         \[\leadsto \color{blue}{\left|\mathsf{neg}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right)\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{-d}}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      9. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{-\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      11. fabs-div N/A

          \[\leadsto \color{blue}{\frac{\left|\mathsf{neg}\left(\left(-d\right)\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\left|\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto \frac{\left|\color{blue}{d}\right|}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      14. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      17. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)}}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(-d\right)}\right)}{\left|-\sqrt{h \cdot \ell}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      20. neg-fabs N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      22. sqrt-fabs-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      23. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

      24. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-d\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)\right)\right)}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{1}{4}\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d + d\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 58.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot 1\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(0.5 \cdot h\right) \cdot t\_2\right) \cdot t\_3}{\left(\ell \cdot d\right) \cdot d} \cdot t\_1, -0.25, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (fabs (/ d (sqrt (* l h)))) 1.0))
        (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 -1e+86)
     (*
      (fma (* (/ (* (* (* 0.5 h) t_2) t_3) (* (* l d) d)) t_1) -0.25 1.0)
      (/ d (sqrt (* h l))))
     (if (<= t_4 0.0)
       t_0
       (if (<= t_4 5e+144)
         (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h)
         t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((l * h)))) * 1.0;
	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 <= -1e+86) {
		tmp = fma((((((0.5 * h) * t_2) * t_3) / ((l * d) * d)) * t_1), -0.25, 1.0) * (d / sqrt((h * l)));
	} else if (t_4 <= 0.0) {
		tmp = t_0;
	} else if (t_4 <= 5e+144) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * 1.0)
	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 <= -1e+86)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(0.5 * h) * t_2) * t_3) / Float64(Float64(l * d) * d)) * t_1), -0.25, 1.0) * Float64(d / sqrt(Float64(h * l))));
	elseif (t_4 <= 0.0)
		tmp = t_0;
	elseif (t_4 <= 5e+144)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $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, -1e+86], N[(N[(N[(N[(N[(N[(N[(0.5 * h), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$0, If[LessEqual[t$95$4, 5e+144], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], t$95$0]]]]]]]]

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)))) < -1e86

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot -0.25, \frac{\left(0.5 \cdot h\right) \cdot D}{\ell \cdot d}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   7. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{D}{\ell \cdot d} \cdot \left(0.5 \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot M\right), -0.25, 1\right) \cdot d}{-\sqrt{h \cdot \ell}}} \]

   9. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(0.5 \cdot h\right) \cdot D\right) \cdot \left(M \cdot D\right)}{\left(\ell \cdot d\right) \cdot d} \cdot M, -0.25, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   if -1e86 < (*.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 4.9999999999999999e144 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]

   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)))) < 4.9999999999999999e144

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 54.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot 1\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-33}:\\ \;\;\;\;-1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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 (* (fabs (/ d (sqrt (* l h)))) 1.0)))
   (if (<= t_0 -1e-33)
     (* -1.0 (/ d (* h (sqrt (/ l h)))))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 5e+144)
         (/ (* h (* (sqrt (/ d h)) (sqrt (/ d l)))) h)
         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 = fabs((d / sqrt((l * h)))) * 1.0;
	double tmp;
	if (t_0 <= -1e-33) {
		tmp = -1.0 * (d / (h * sqrt((l / h))));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+144) {
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	} 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 = abs((d / sqrt((l * h)))) * 1.0d0
    if (t_0 <= (-1d-33)) then
        tmp = (-1.0d0) * (d / (h * sqrt((l / h))))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+144) then
        tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h
    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.abs((d / Math.sqrt((l * h)))) * 1.0;
	double tmp;
	if (t_0 <= -1e-33) {
		tmp = -1.0 * (d / (h * Math.sqrt((l / h))));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+144) {
		tmp = (h * (Math.sqrt((d / h)) * Math.sqrt((d / l)))) / h;
	} 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.fabs((d / math.sqrt((l * h)))) * 1.0
	tmp = 0
	if t_0 <= -1e-33:
		tmp = -1.0 * (d / (h * math.sqrt((l / h))))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+144:
		tmp = (h * (math.sqrt((d / h)) * math.sqrt((d / l)))) / h
	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(abs(Float64(d / sqrt(Float64(l * h)))) * 1.0)
	tmp = 0.0
	if (t_0 <= -1e-33)
		tmp = Float64(-1.0 * Float64(d / Float64(h * sqrt(Float64(l / h)))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+144)
		tmp = Float64(Float64(h * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))) / h);
	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 = abs((d / sqrt((l * h)))) * 1.0;
	tmp = 0.0;
	if (t_0 <= -1e-33)
		tmp = -1.0 * (d / (h * sqrt((l / h))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+144)
		tmp = (h * (sqrt((d / h)) * sqrt((d / l)))) / h;
	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[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-33], N[(-1.0 * N[(d / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+144], N[(N[(h * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $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)))) < -1.0000000000000001e-33

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.9%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.7%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   7. Taylor expanded in h around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

      5. lower-/.f64 12.1%

         \[\leadsto -1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

   9. Applied rewrites 12.1%

      \[\leadsto -1 \cdot \color{blue}{\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}} \]

   if -1.0000000000000001e-33 < (*.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 4.9999999999999999e144 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]

   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)))) < 4.9999999999999999e144

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 24.0%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

      6. lower-/.f64 38.2%

         \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]

   7. Applied rewrites 38.2%

      \[\leadsto \frac{h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 49.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-33}:\\ \;\;\;\;-1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -1e-33)
   (* -1.0 (/ d (* h (sqrt (/ l h)))))
   (* (fabs (/ d (sqrt (* l h)))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-33) {
		tmp = -1.0 * (d / (h * sqrt((l / h))));
	} else {
		tmp = fabs((d / sqrt((l * h)))) * 1.0;
	}
	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 (((((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)))) <= (-1d-33)) then
        tmp = (-1.0d0) * (d / (h * sqrt((l / h))))
    else
        tmp = abs((d / sqrt((l * h)))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-33) {
		tmp = -1.0 * (d / (h * Math.sqrt((l / h))));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h)))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-33:
		tmp = -1.0 * (d / (h * math.sqrt((l / h))))
	else:
		tmp = math.fabs((d / math.sqrt((l * h)))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-33)
		tmp = Float64(-1.0 * Float64(d / Float64(h * sqrt(Float64(l / h)))));
	else
		tmp = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -1e-33)
		tmp = -1.0 * (d / (h * sqrt((l / h))));
	else
		tmp = abs((d / sqrt((l * h)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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], -1e-33], N[(-1.0 * N[(d / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.0000000000000001e-33

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.9%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.7%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   7. Taylor expanded in h around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

      5. lower-/.f64 12.1%

         \[\leadsto -1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

   9. Applied rewrites 12.1%

      \[\leadsto -1 \cdot \color{blue}{\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}} \]

   if -1.0000000000000001e-33 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 47.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\left(h \cdot \frac{d}{\ell}\right) \cdot d}}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-5)
   (/ (sqrt (* (* h (/ d l)) d)) h)
   (* (fabs (/ d (sqrt (* l h)))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-5) {
		tmp = sqrt(((h * (d / l)) * d)) / h;
	} else {
		tmp = fabs((d / sqrt((l * h)))) * 1.0;
	}
	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 (((((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)))) <= (-5d-5)) then
        tmp = sqrt(((h * (d / l)) * d)) / h
    else
        tmp = abs((d / sqrt((l * h)))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-5) {
		tmp = Math.sqrt(((h * (d / l)) * d)) / h;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h)))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-5:
		tmp = math.sqrt(((h * (d / l)) * d)) / h
	else:
		tmp = math.fabs((d / math.sqrt((l * h)))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-5)
		tmp = Float64(sqrt(Float64(Float64(h * Float64(d / l)) * d)) / h);
	else
		tmp = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-5)
		tmp = sqrt(((h * (d / l)) * d)) / h;
	else
		tmp = abs((d / sqrt((l * h)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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], -5e-5], N[(N[Sqrt[N[(N[(h * N[(d / l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision], N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000024e-5

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{-d}{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. sqrt-undiv N/A

          \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{-d}}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{-h}}} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{-h}}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{-h}}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. pow1/2 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{\color{blue}{\sqrt{\frac{d}{\ell}}}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{-h}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. sqrt-undiv N/A

          \[\leadsto \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{-h}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{-h}}}\right) \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 29.7%

      \[\leadsto \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{\frac{d}{\ell}}{-h}}\right)} \cdot \left(1 - \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

      8. lower-*.f64 5.9%

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h} \]

   8. Applied rewrites 5.9%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \sqrt{-1 \cdot \frac{d \cdot h}{\ell}}}{h}} \]
   9. Step-by-step derivation

   10. Applied rewrites 22.3%

       \[\leadsto \frac{\sqrt{\left(h \cdot \frac{d}{\ell}\right) \cdot d}}{\color{blue}{h}} \]

   if -5.00000000000000024e-5 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 46.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -1e-33)
   (/ (- d) (sqrt (* h l)))
   (* (fabs (/ d (sqrt (* l h)))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-33) {
		tmp = -d / sqrt((h * l));
	} else {
		tmp = fabs((d / sqrt((l * h)))) * 1.0;
	}
	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 (((((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)))) <= (-1d-33)) then
        tmp = -d / sqrt((h * l))
    else
        tmp = abs((d / sqrt((l * h)))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-33) {
		tmp = -d / Math.sqrt((h * l));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h)))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-33:
		tmp = -d / math.sqrt((h * l))
	else:
		tmp = math.fabs((d / math.sqrt((l * h)))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-33)
		tmp = Float64(Float64(-d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -1e-33)
		tmp = -d / sqrt((h * l));
	else
		tmp = abs((d / sqrt((l * h)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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], -1e-33], N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.0000000000000001e-33

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.9%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.7%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(d\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{-d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      6. lower-/.f64 26.7%

         \[\leadsto \frac{-d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   if -1.0000000000000001e-33 < (*.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 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)} \cdot \frac{h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{\color{blue}{d + d}} \cdot M\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

   3. Applied rewrites 68.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      11. frac-times N/A

          \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      14. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      15. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      17. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      18. sqrt-fabs-rev N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\left|\sqrt{h \cdot \ell}\right|}} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      19. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      20. div-fabs N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      21. lower-fabs.f64 N/A

          \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      22. lower-/.f64 71.3%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot \frac{1}{2}\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

      25. lower-*.f64 71.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 43.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-254}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l)))) (if (<= l -2.1e-254) (/ (- d) t_0) (/ d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -2.1e-254) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	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((h * l))
    if (l <= (-2.1d-254)) then
        tmp = -d / t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (l <= -2.1e-254) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if l <= -2.1e-254:
		tmp = -d / t_0
	else:
		tmp = d / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -2.1e-254)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (l <= -2.1e-254)
		tmp = -d / t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.1e-254], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.09999999999999997e-254

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.9%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.7%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(d\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{-d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      6. lower-/.f64 26.7%

         \[\leadsto \frac{-d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   if -2.09999999999999997e-254 < l

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.9%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.7%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(d\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{-d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      6. lower-/.f64 26.7%

         \[\leadsto \frac{-d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-d}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(-d\right) \cdot \left(-d\right)}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(-d\right)}}{\sqrt{h \cdot \ell}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{h \cdot \ell}} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h} \cdot \ell}} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      7. rem-square-sqrt 26.6%

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied rewrites 26.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 26.6% accurate, 10.2× speedup

Math

\[\frac{d}{\sqrt{h \cdot \ell}} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((h * l));
}

Fortran

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))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. unpow1/2 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. frac-times N/A

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. sqrt-div N/A

       \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. lower-unsound-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lower-unsound-sqrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   15. lower-unsound-sqrt.f64 N/A

       \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   16. lower-*.f64 48.9%

       \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 48.9%

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   4. lower-*.f64 26.7%

      \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

6. Applied rewrites 26.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   2. lift-/.f64 N/A

      \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(d\right)}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{-d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   6. lower-/.f64 26.7%

      \[\leadsto \frac{-d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

8. Applied rewrites 26.7%

   \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \frac{\sqrt{-d} \cdot \sqrt{-d}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   2. sqrt-unprod N/A

      \[\leadsto \frac{\sqrt{\left(-d\right) \cdot \left(-d\right)}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(-d\right)}}{\sqrt{h \cdot \ell}} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{h \cdot \ell}} \]

   5. sqr-neg-rev N/A

      \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h} \cdot \ell}} \]

   6. sqrt-unprod N/A

      \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   7. rem-square-sqrt 26.6%

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

10. Applied rewrites 26.6%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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)))))