Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 85.2%

Time: 10.2s

Alternatives: 17

Speedup: 1.0×

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: 65.8% 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: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-0.5}{d} \cdot \left(D \cdot M\right)\\ t_1 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := t\_1 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;t\_1 \cdot \left(1 - t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+234}:\\ \;\;\;\;t\_1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (/ -0.5 d) (* D M)))
       (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
        (*
         (/ (fabs d) (sqrt (* h l)))
         (-
          1.0
          (*
           (* (* M D) (/ 0.25 d))
           (/ (* (* D M) h) (* (+ d d) l)))))))
  (if (<= t_2 -1e-147)
    (* t_1 (- 1.0 (* t_0 (* t_0 (* 0.5 (/ h l))))))
    (if (<= t_2 0.0) t_3 (if (<= t_2 1e+234) (* t_1 1.0) t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (-0.5 / d) * (D * M);
	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 = (fabs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))));
	double tmp;
	if (t_2 <= -1e-147) {
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 1e+234) {
		tmp = t_1 * 1.0;
	} else {
		tmp = 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) :: tmp
    t_0 = ((-0.5d0) / d) * (d_1 * m)
    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 = (abs(d) / sqrt((h * l))) * (1.0d0 - (((m * d_1) * (0.25d0 / d)) * (((d_1 * m) * h) / ((d + d) * l))))
    if (t_2 <= (-1d-147)) then
        tmp = t_1 * (1.0d0 - (t_0 * (t_0 * (0.5d0 * (h / l)))))
    else if (t_2 <= 0.0d0) then
        tmp = t_3
    else if (t_2 <= 1d+234) then
        tmp = t_1 * 1.0d0
    else
        tmp = 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 = (-0.5 / d) * (D * M);
	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 = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))));
	double tmp;
	if (t_2 <= -1e-147) {
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 1e+234) {
		tmp = t_1 * 1.0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (-0.5 / d) * (D * M)
	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 = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))))
	tmp = 0
	if t_2 <= -1e-147:
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))))
	elif t_2 <= 0.0:
		tmp = t_3
	elif t_2 <= 1e+234:
		tmp = t_1 * 1.0
	else:
		tmp = t_3
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(-0.5 / d) * Float64(D * M))
	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(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(0.25 / d)) * Float64(Float64(Float64(D * M) * h) / Float64(Float64(d + d) * l)))))
	tmp = 0.0
	if (t_2 <= -1e-147)
		tmp = Float64(t_1 * Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(0.5 * Float64(h / l))))));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 1e+234)
		tmp = Float64(t_1 * 1.0);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (-0.5 / d) * (D * M);
	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 = (abs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))));
	tmp = 0.0;
	if (t_2 <= -1e-147)
		tmp = t_1 * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 1e+234)
		tmp = t_1 * 1.0;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(-0.5 / d), $MachinePrecision] * N[(D * M), $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[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-147], N[(t$95$1 * N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 1e+234], N[(t$95$1 * 1.0), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999997e-148

   1. Initial program 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. sqr-neg-rev N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 67.4%

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

   if -9.9999999999999997e-148 < (*.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 1e234 < (*.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 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 74.7%

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

   7. Applied rewrites 74.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-fabs-rev N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 74.7%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 74.7%

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

   9. Applied rewrites 74.7%

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

   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)))) < 1e234

   1. Initial program 65.8%

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

   2. 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 38.8%

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

Alternative 2: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(M, D\right) \cdot \mathsf{min}\left(M, D\right)\\ t_1 := \frac{t\_0 \cdot h}{\left(d + d\right) \cdot \ell}\\ t_2 := \frac{\mathsf{max}\left(M, D\right)}{d + d} \cdot \mathsf{min}\left(M, D\right)\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+91}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 3 \cdot 10^{-297}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot \frac{0.25}{d}\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell} \cdot \sqrt{h}}\right| \cdot \left(1 - \frac{0.5 \cdot t\_0}{d + d} \cdot t\_1\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (fmax M D) (fmin M D)))
       (t_1 (/ (* t_0 h) (* (+ d d) l)))
       (t_2 (* (/ (fmax M D) (+ d d)) (fmin M D))))
  (if (<= h -1.5e+91)
    (*
     (* (/ (sqrt (- d)) (sqrt (- h))) (pow (/ d l) (/ 1.0 2.0)))
     (- 1.0 (* (* (/ 1.0 2.0) (* t_2 t_2)) (/ h l))))
    (if (<= h 3e-297)
      (*
       (/ (fabs d) (sqrt (* h l)))
       (- 1.0 (* (* (* (fmin M D) (fmax M D)) (/ 0.25 d)) t_1)))
      (*
       (fabs (/ (- d) (* (sqrt l) (sqrt h))))
       (- 1.0 (* (/ (* 0.5 t_0) (+ d d)) t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(M, D) * fmin(M, D);
	double t_1 = (t_0 * h) / ((d + d) * l);
	double t_2 = (fmax(M, D) / (d + d)) * fmin(M, D);
	double tmp;
	if (h <= -1.5e+91) {
		tmp = ((sqrt(-d) / sqrt(-h)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (t_2 * t_2)) * (h / l)));
	} else if (h <= 3e-297) {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - (((fmin(M, D) * fmax(M, D)) * (0.25 / d)) * t_1));
	} else {
		tmp = fabs((-d / (sqrt(l) * sqrt(h)))) * (1.0 - (((0.5 * t_0) / (d + 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 = fmax(m, d_1) * fmin(m, d_1)
    t_1 = (t_0 * h) / ((d + d) * l)
    t_2 = (fmax(m, d_1) / (d + d)) * fmin(m, d_1)
    if (h <= (-1.5d+91)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (t_2 * t_2)) * (h / l)))
    else if (h <= 3d-297) then
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - (((fmin(m, d_1) * fmax(m, d_1)) * (0.25d0 / d)) * t_1))
    else
        tmp = abs((-d / (sqrt(l) * sqrt(h)))) * (1.0d0 - (((0.5d0 * t_0) / (d + d)) * 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 = fmax(M, D) * fmin(M, D);
	double t_1 = (t_0 * h) / ((d + d) * l);
	double t_2 = (fmax(M, D) / (d + d)) * fmin(M, D);
	double tmp;
	if (h <= -1.5e+91) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (t_2 * t_2)) * (h / l)));
	} else if (h <= 3e-297) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (((fmin(M, D) * fmax(M, D)) * (0.25 / d)) * t_1));
	} else {
		tmp = Math.abs((-d / (Math.sqrt(l) * Math.sqrt(h)))) * (1.0 - (((0.5 * t_0) / (d + d)) * t_1));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmax(M, D) * fmin(M, D)
	t_1 = (t_0 * h) / ((d + d) * l)
	t_2 = (fmax(M, D) / (d + d)) * fmin(M, D)
	tmp = 0
	if h <= -1.5e+91:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (t_2 * t_2)) * (h / l)))
	elif h <= 3e-297:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (((fmin(M, D) * fmax(M, D)) * (0.25 / d)) * t_1))
	else:
		tmp = math.fabs((-d / (math.sqrt(l) * math.sqrt(h)))) * (1.0 - (((0.5 * t_0) / (d + d)) * t_1))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(fmax(M, D) * fmin(M, D))
	t_1 = Float64(Float64(t_0 * h) / Float64(Float64(d + d) * l))
	t_2 = Float64(Float64(fmax(M, D) / Float64(d + d)) * fmin(M, D))
	tmp = 0.0
	if (h <= -1.5e+91)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(t_2 * t_2)) * Float64(h / l))));
	elseif (h <= 3e-297)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(fmin(M, D) * fmax(M, D)) * Float64(0.25 / d)) * t_1)));
	else
		tmp = Float64(abs(Float64(Float64(-d) / Float64(sqrt(l) * sqrt(h)))) * Float64(1.0 - Float64(Float64(Float64(0.5 * t_0) / Float64(d + d)) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = max(M, D) * min(M, D);
	t_1 = (t_0 * h) / ((d + d) * l);
	t_2 = (max(M, D) / (d + d)) * min(M, D);
	tmp = 0.0;
	if (h <= -1.5e+91)
		tmp = ((sqrt(-d) / sqrt(-h)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (t_2 * t_2)) * (h / l)));
	elseif (h <= 3e-297)
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - (((min(M, D) * max(M, D)) * (0.25 / d)) * t_1));
	else
		tmp = abs((-d / (sqrt(l) * sqrt(h)))) * (1.0 - (((0.5 * t_0) / (d + d)) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Max[M, D], $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Max[M, D], $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.5e+91], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3e-297], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.5e91

   1. Initial program 65.8%

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

      1. lift-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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 37.7%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 37.3%

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 37.5%

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

   5. Applied rewrites 37.5%

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

   if -1.5e91 < h < 2.9999999999999999e-297

   1. Initial program 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 74.7%

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

   7. Applied rewrites 74.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-fabs-rev N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 74.7%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 74.7%

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

   9. Applied rewrites 74.7%

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

   if 2.9999999999999999e-297 < h

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 41.6%

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

   7. Applied rewrites 41.6%

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

Alternative 3: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_1 := t\_0 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+234}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right) \cdot t\_2\right)\\ \end{array} \]

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))))
       (t_1
        (*
         t_0
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_2 (/ (* (* D M) h) (* (+ d d) l))))
  (if (<= t_1 0.0)
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (- 1.0 (* (* (/ (* M D) d) 0.25) t_2)))
    (if (<= t_1 1e+234)
      (* t_0 1.0)
      (*
       (/ (fabs d) (sqrt (* h l)))
       (- 1.0 (* (* (* M D) (/ 0.25 d)) t_2)))))))

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));
	double t_1 = t_0 * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = ((D * M) * h) / ((d + d) * l);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_2));
	} else if (t_1 <= 1e+234) {
		tmp = t_0 * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / 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) :: tmp
    t_0 = ((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))
    t_1 = t_0 * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_2 = ((d_1 * m) * h) / ((d + d) * l)
    if (t_1 <= 0.0d0) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - ((((m * d_1) / d) * 0.25d0) * t_2))
    else if (t_1 <= 1d+234) then
        tmp = t_0 * 1.0d0
    else
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - (((m * d_1) * (0.25d0 / d)) * 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 = Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0));
	double t_1 = t_0 * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = ((D * M) * h) / ((d + d) * l);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_2));
	} else if (t_1 <= 1e+234) {
		tmp = t_0 * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * t_2));
	}
	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))
	t_1 = t_0 * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = ((D * M) * h) / ((d + d) * l)
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_2))
	elif t_1 <= 1e+234:
		tmp = t_0 * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * t_2))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))
	t_1 = Float64(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_2 = Float64(Float64(Float64(D * M) * h) / Float64(Float64(d + d) * l))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(Float64(M * D) / d) * 0.25) * t_2)));
	elseif (t_1 <= 1e+234)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(0.25 / d)) * t_2)));
	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));
	t_1 = t_0 * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = ((D * M) * h) / ((d + d) * l);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_2));
	elseif (t_1 <= 1e+234)
		tmp = t_0 * 1.0;
	else
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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$1 = N[(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]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+234], N[(t$95$0 * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(0.25 / 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)))) < 0.0

   1. Initial program 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 74.7%

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

   7. Applied rewrites 74.7%

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

   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)))) < 1e234

   1. Initial program 65.8%

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

   2. 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 38.8%

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

   if 1e234 < (*.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 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 74.7%

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

   7. Applied rewrites 74.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-fabs-rev N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 74.7%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 74.7%

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

   9. Applied rewrites 74.7%

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

Alternative 4: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot t\_1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+234}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right) \cdot t\_1\right)\\ \end{array} \]

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 (/ (* (* D M) h) (* (+ d d) l))))
  (if (<= t_0 0.0)
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (- 1.0 (* (* (/ (* M D) d) 0.25) t_1)))
    (if (<= t_0 1e+234)
      (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
      (*
       (/ (fabs d) (sqrt (* h l)))
       (- 1.0 (* (* (* M D) (/ 0.25 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 = ((D * M) * h) / ((d + d) * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_1));
	} else if (t_0 <= 1e+234) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / 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) :: 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) / ((d + d) * l)
    if (t_0 <= 0.0d0) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - ((((m * d_1) / d) * 0.25d0) * t_1))
    else if (t_0 <= 1d+234) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - (((m * d_1) * (0.25d0 / d)) * 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 = ((D * M) * h) / ((d + d) * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_1));
	} else if (t_0 <= 1e+234) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / 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 = ((D * M) * h) / ((d + d) * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_1))
	elif t_0 <= 1e+234:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / 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(Float64(Float64(D * M) * h) / Float64(Float64(d + d) * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(Float64(M * D) / d) * 0.25) * t_1)));
	elseif (t_0 <= 1e+234)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(0.25 / 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 = ((D * M) * h) / ((d + d) * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - ((((M * D) / d) * 0.25) * t_1));
	elseif (t_0 <= 1e+234)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / 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[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+234], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(0.25 / 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 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 74.7%

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

   7. Applied rewrites 74.7%

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

   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)))) < 1e234

   1. Initial program 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

   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 43.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. times-frac N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

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

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

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

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

      23. lower-/.f64 38.8%

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

   10. Applied rewrites 38.8%

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

   if 1e234 < (*.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 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 74.7%

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

   7. Applied rewrites 74.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-fabs-rev N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 74.7%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 74.7%

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

   9. Applied rewrites 74.7%

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

Alternative 5: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+234}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1
        (*
         (/ (fabs d) (sqrt (* h l)))
         (-
          1.0
          (*
           (* (* M D) (/ 0.25 d))
           (/ (* (* D M) h) (* (+ d d) l)))))))
  (if (<= t_0 0.0)
    t_1
    (if (<= t_0 1e+234)
      (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
      t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+234) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (abs(d) / sqrt((h * l))) * (1.0d0 - (((m * d_1) * (0.25d0 / d)) * (((d_1 * m) * h) / ((d + d) * l))))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+234) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+234) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+234:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(0.25 / d)) * Float64(Float64(Float64(D * M) * h) / Float64(Float64(d + d) * l)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+234)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (abs(d) / sqrt((h * l))) * (1.0 - (((M * D) * (0.25 / d)) * (((D * M) * h) / ((d + d) * l))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+234)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+234], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 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 1e234 < (*.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 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 74.7%

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

   7. Applied rewrites 74.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-fabs-rev N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 74.7%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 74.7%

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

   9. Applied rewrites 74.7%

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

   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)))) < 1e234

   1. Initial program 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

   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 43.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. times-frac N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-prod N/A

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

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

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

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

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

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

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

      23. lower-/.f64 38.8%

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

   10. Applied rewrites 38.8%

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

Alternative 6: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(D \cdot M\right) \cdot h\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \left(d + d\right) \cdot \ell\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(\frac{0.25 \cdot t\_0}{d \cdot t\_3}, D \cdot M, -1\right) \cdot \frac{\left|d\right|}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{t\_0 \cdot \left(\left(D \cdot M\right) \cdot 0.25\right)}{t\_3 \cdot d}\right) \cdot \left|d\right|}{t\_1}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (* D M) h))
       (t_1 (sqrt (* h l)))
       (t_2
        (*
         (* (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_3 (* (+ d d) l)))
  (if (<= t_2 0.0)
    (-
     (*
      (fma (/ (* 0.25 t_0) (* d t_3)) (* D M) -1.0)
      (/ (fabs d) t_1)))
    (if (<= t_2 2e+256)
      (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
      (/
       (* (- 1.0 (/ (* t_0 (* (* D M) 0.25)) (* t_3 d))) (fabs d))
       t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) * h;
	double t_1 = sqrt((h * l));
	double t_2 = (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_3 = (d + d) * l;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -(fma(((0.25 * t_0) / (d * t_3)), (D * M), -1.0) * (fabs(d) / t_1));
	} else if (t_2 <= 2e+256) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = ((1.0 - ((t_0 * ((D * M) * 0.25)) / (t_3 * d))) * fabs(d)) / t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) * h)
	t_1 = sqrt(Float64(h * l))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(Float64(d + d) * l)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-Float64(fma(Float64(Float64(0.25 * t_0) / Float64(d * t_3)), Float64(D * M), -1.0) * Float64(abs(d) / t_1)));
	elseif (t_2 <= 2e+256)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(t_0 * Float64(Float64(D * M) * 0.25)) / Float64(t_3 * d))) * abs(d)) / t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(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[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], (-N[(N[(N[(N[(0.25 * t$95$0), $MachinePrecision] / N[(d * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$2, 2e+256], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(t$95$0 * N[(N[(D * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / t$95$1), $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 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 74.7%

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

   7. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 68.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e256

   1. Initial program 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. pow-prod-down N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fabs.f64 65.2%

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

   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 43.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-fabs N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-fabs-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot 1 \]

      18. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot 1 \]

      19. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      22. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot 1 \]

      23. lower-/.f64 38.8%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot 1 \]

   10. Applied rewrites 38.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

   if 2.0000000000000001e256 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(D \cdot M\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d + d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. count-2 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. times-frac N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{\frac{1}{2}}{2} \cdot \frac{M \cdot D}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\frac{1}{4}} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\frac{1}{4} \cdot \left(M \cdot D\right)}{d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{4}}}{d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. associate-/l* N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lower-/.f64 74.7%

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{0.25}{d}}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   7. Applied rewrites 74.7%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \]

      5. fabs-div N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \]

      6. lift-neg.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \]

      7. neg-fabs N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

   9. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{\left(\left(D \cdot M\right) \cdot h\right) \cdot \left(\left(D \cdot M\right) \cdot 0.25\right)}{\left(\left(d + d\right) \cdot \ell\right) \cdot d}\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left(1 - \frac{\left(\left(D \cdot M\right) \cdot h\right) \cdot \left(\left(D \cdot M\right) \cdot 0.25\right)}{\left(\left(d + d\right) \cdot \ell\right) \cdot d}\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1
        (/
         (*
          (-
           1.0
           (/ (* (* (* D M) h) (* (* D M) 0.25)) (* (* (+ d d) l) d)))
          (fabs d))
         (sqrt (* h l)))))
  (if (<= t_0 0.0)
    t_1
    (if (<= t_0 2e+256)
      (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
      t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = ((1.0 - ((((D * M) * h) * ((D * M) * 0.25)) / (((d + d) * l) * d))) * fabs(d)) / sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+256) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = ((1.0d0 - ((((d_1 * m) * h) * ((d_1 * m) * 0.25d0)) / (((d + d) * l) * d))) * abs(d)) / sqrt((h * l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+256) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = ((1.0 - ((((D * M) * h) * ((D * M) * 0.25)) / (((d + d) * l) * d))) * Math.abs(d)) / Math.sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+256) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = ((1.0 - ((((D * M) * h) * ((D * M) * 0.25)) / (((d + d) * l) * d))) * math.fabs(d)) / math.sqrt((h * l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+256:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(D * M) * h) * Float64(Float64(D * M) * 0.25)) / Float64(Float64(Float64(d + d) * l) * d))) * abs(d)) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+256)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = ((1.0 - ((((D * M) * h) * ((D * M) * 0.25)) / (((d + d) * l) * d))) * abs(d)) / sqrt((h * l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+256)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 - N[(N[(N[(N[(D * M), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+256], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 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 2.0000000000000001e256 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(D \cdot M\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d + d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. count-2 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. times-frac N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\frac{\frac{1}{2}}{2} \cdot \frac{M \cdot D}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\color{blue}{\frac{1}{4}} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\frac{\frac{1}{4} \cdot \left(M \cdot D\right)}{d}} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{4}}}{d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. associate-/l* N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lower-/.f64 74.7%

          \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{0.25}{d}}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   7. Applied rewrites 74.7%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right)} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \]

      5. fabs-div N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \]

      6. lift-neg.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \]

      7. neg-fabs N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h}}} \]

   9. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{\left(\left(D \cdot M\right) \cdot h\right) \cdot \left(\left(D \cdot M\right) \cdot 0.25\right)}{\left(\left(d + d\right) \cdot \ell\right) \cdot d}\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e256

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. fabs-div N/A

         \[\leadsto \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      5. neg-fabs N/A

         \[\leadsto \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      13. sqrt-undiv N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot 1 \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      17. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot 1 \]

      18. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot 1 \]

      19. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      22. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot 1 \]

      23. lower-/.f64 38.8%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot 1 \]

   10. Applied rewrites 38.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := \mathsf{fma}\left(\left(\left(\left(\left(\mathsf{max}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot h\right) \cdot 0.25\right) \cdot \mathsf{min}\left(M, D\right)\right) \cdot \frac{\mathsf{min}\left(M, D\right)}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \frac{\left|d\right|}{t\_1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|d \cdot \frac{-1}{t\_1}\right| \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+234}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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 (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1 (sqrt (* h l)))
       (t_2
        (*
         (fma
          (*
           (* (* (* (* (fmax M D) (fmax M D)) h) 0.25) (fmin M D))
           (/ (fmin M D) (* (* d d) l)))
          -0.5
          1.0)
         (/ (fabs d) t_1))))
  (if (<= t_0 -2e-45)
    t_2
    (if (<= t_0 0.0)
      (* (fabs (* d (/ -1.0 t_1))) 1.0)
      (if (<= t_0 1e+234)
        (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
        (if (<= t_0 INFINITY)
          (* (* (sqrt (/ 1.0 (* h l))) (fabs d)) 1.0)
          t_2))))))

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(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((h * l));
	double t_2 = fma((((((fmax(M, D) * fmax(M, D)) * h) * 0.25) * fmin(M, D)) * (fmin(M, D) / ((d * d) * l))), -0.5, 1.0) * (fabs(d) / t_1);
	double tmp;
	if (t_0 <= -2e-45) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d * (-1.0 / t_1))) * 1.0;
	} else if (t_0 <= 1e+234) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((1.0 / (h * l))) * fabs(d)) * 1.0;
	} else {
		tmp = t_2;
	}
	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(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(h * l))
	t_2 = Float64(fma(Float64(Float64(Float64(Float64(Float64(fmax(M, D) * fmax(M, D)) * h) * 0.25) * fmin(M, D)) * Float64(fmin(M, D) / Float64(Float64(d * d) * l))), -0.5, 1.0) * Float64(abs(d) / t_1))
	tmp = 0.0
	if (t_0 <= -2e-45)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(abs(Float64(d * Float64(-1.0 / t_1))) * 1.0);
	elseif (t_0 <= 1e+234)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * abs(d)) * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(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]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * 0.25), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[(N[Min[M, D], $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-45], t$95$2, If[LessEqual[t$95$0, 0.0], N[(N[Abs[N[(d * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+234], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$2]]]]]]]

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)))) < -2e-45 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{\color{blue}{h \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{\color{blue}{\ell \cdot h}}} \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \color{blue}{\frac{\frac{d}{\ell}}{h}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{\color{blue}{\frac{d}{\ell}}}{h}} \]

      6. lower-/.f64 37.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \color{blue}{\frac{\frac{d}{\ell}}{h}}} \]

   5. Applied rewrites 37.3%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \color{blue}{\frac{\frac{d}{\ell}}{h}}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot 0.25\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if -2e-45 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      2. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\left(-d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left|\color{blue}{d} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{d \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left|d \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      8. frac-2neg-rev N/A

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      9. lower-/.f64 43.0%

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

      12. lift-*.f64 43.0%

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

   10. Applied rewrites 43.0%

       \[\leadsto \left|\color{blue}{d \cdot \frac{-1}{\sqrt{h \cdot \ell}}}\right| \cdot 1 \]

   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)))) < 1e234

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. fabs-div N/A

         \[\leadsto \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      5. neg-fabs N/A

         \[\leadsto \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      13. sqrt-undiv N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot 1 \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      17. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot 1 \]

      18. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot 1 \]

      19. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      22. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot 1 \]

      23. lower-/.f64 38.8%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot 1 \]

   10. Applied rewrites 38.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

   if 1e234 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(-d\right)}\right| \cdot 1 \]

      5. fabs-mul N/A

         \[\leadsto \color{blue}{\left(\left|\frac{1}{\sqrt{\ell \cdot h}}\right| \cdot \left|-d\right|\right)} \cdot 1 \]

      6. div-fabs N/A

         \[\leadsto \left(\color{blue}{\frac{\left|1\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot \left|-d\right|\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\left|\sqrt{\ell \cdot h}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|\color{blue}{\mathsf{neg}\left(d\right)}\right|\right) \cdot 1 \]

      12. neg-fabs N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left|d\right|}\right) \cdot 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|d\right|\right)} \cdot 1 \]

   10. Applied rewrites 42.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right)} \cdot 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\right| \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+234}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1
        (*
         (fma
          (* -0.5 h)
          (/ (* (* (* D D) 0.25) (* M M)) (* (* d d) l))
          1.0)
         (/ (fabs d) (sqrt (* l h))))))
  (if (<= t_0 -2e-45)
    t_1
    (if (<= t_0 0.0)
      (* (fabs (* d (/ -1.0 (sqrt (* h l))))) 1.0)
      (if (<= t_0 1e+234)
        (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
        (if (<= t_0 INFINITY)
          (* (* (sqrt (/ 1.0 (* h l))) (fabs d)) 1.0)
          t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fma((-0.5 * h), ((((D * D) * 0.25) * (M * M)) / ((d * d) * l)), 1.0) * (fabs(d) / sqrt((l * h)));
	double tmp;
	if (t_0 <= -2e-45) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = fabs((d * (-1.0 / sqrt((h * l))))) * 1.0;
	} else if (t_0 <= 1e+234) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((1.0 / (h * l))) * fabs(d)) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(fma(Float64(-0.5 * h), Float64(Float64(Float64(Float64(D * D) * 0.25) * Float64(M * M)) / Float64(Float64(d * d) * l)), 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= -2e-45)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(abs(Float64(d * Float64(-1.0 / sqrt(Float64(h * l))))) * 1.0);
	elseif (t_0 <= 1e+234)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * abs(d)) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-45], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[Abs[N[(d * N[(-1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+234], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]]]

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)))) < -2e-45 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{\sqrt{\ell \cdot h}} \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{\sqrt{\ell \cdot h}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|-d\right|}}{\sqrt{\ell \cdot h}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\sqrt{\ell \cdot h}} \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

      6. lower-fabs.f64 49.4%

         \[\leadsto \mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 49.4%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \]

   if -2e-45 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      2. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\left(-d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left|\color{blue}{d} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{d \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left|d \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      8. frac-2neg-rev N/A

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      9. lower-/.f64 43.0%

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

      12. lift-*.f64 43.0%

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

   10. Applied rewrites 43.0%

       \[\leadsto \left|\color{blue}{d \cdot \frac{-1}{\sqrt{h \cdot \ell}}}\right| \cdot 1 \]

   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)))) < 1e234

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. fabs-div N/A

         \[\leadsto \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      5. neg-fabs N/A

         \[\leadsto \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      13. sqrt-undiv N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot 1 \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      17. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot 1 \]

      18. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot 1 \]

      19. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      22. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot 1 \]

      23. lower-/.f64 38.8%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot 1 \]

   10. Applied rewrites 38.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

   if 1e234 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(-d\right)}\right| \cdot 1 \]

      5. fabs-mul N/A

         \[\leadsto \color{blue}{\left(\left|\frac{1}{\sqrt{\ell \cdot h}}\right| \cdot \left|-d\right|\right)} \cdot 1 \]

      6. div-fabs N/A

         \[\leadsto \left(\color{blue}{\frac{\left|1\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot \left|-d\right|\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\left|\sqrt{\ell \cdot h}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|\color{blue}{\mathsf{neg}\left(d\right)}\right|\right) \cdot 1 \]

      12. neg-fabs N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left|d\right|}\right) \cdot 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|d\right|\right)} \cdot 1 \]

   10. Applied rewrites 42.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right)} \cdot 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+95}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot h, t\_1 \cdot \left(\left(0.25 \cdot t\_1\right) \cdot \left(t\_0 \cdot \frac{t\_0}{\left(d \cdot d\right) \cdot \ell}\right)\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell} \cdot \sqrt{h}}\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) (fabs D))) (t_1 (fmax (fabs M) (fabs D))))
  (if (<= l -9.2e+95)
    (* -1.0 (/ (* d (sqrt (/ -1.0 l))) (sqrt (- h))))
    (if (<= l 1.85e-303)
      (*
       (fma
        (* -0.5 h)
        (* t_1 (* (* 0.25 t_1) (* t_0 (/ t_0 (* (* d d) l)))))
        1.0)
       (/ (- d) (sqrt (* l h))))
      (* (fabs (/ (- d) (* (sqrt l) (sqrt h)))) 1.0)))))

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 tmp;
	if (l <= -9.2e+95) {
		tmp = -1.0 * ((d * sqrt((-1.0 / l))) / sqrt(-h));
	} else if (l <= 1.85e-303) {
		tmp = fma((-0.5 * h), (t_1 * ((0.25 * t_1) * (t_0 * (t_0 / ((d * d) * l))))), 1.0) * (-d / sqrt((l * h)));
	} else {
		tmp = fabs((-d / (sqrt(l) * sqrt(h)))) * 1.0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	tmp = 0.0
	if (l <= -9.2e+95)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(-1.0 / l))) / sqrt(Float64(-h))));
	elseif (l <= 1.85e-303)
		tmp = Float64(fma(Float64(-0.5 * h), Float64(t_1 * Float64(Float64(0.25 * t_1) * Float64(t_0 * Float64(t_0 / Float64(Float64(d * d) * l))))), 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(abs(Float64(Float64(-d) / Float64(sqrt(l) * sqrt(h)))) * 1.0);
	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]}, If[LessEqual[l, -9.2e+95], N[(-1.0 * N[(N[(d * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.85e-303], N[(N[(N[(-0.5 * h), $MachinePrecision] * N[(t$95$1 * N[(N[(0.25 * t$95$1), $MachinePrecision] * N[(t$95$0 * N[(t$95$0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.1999999999999999e95

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. lower-neg.f64 24.1%

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}} \]

   8. Applied rewrites 24.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}} \]

   if -9.1999999999999999e95 < l < 1.8500000000000001e-303

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right)}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right)}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \color{blue}{\left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right)}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\left(\frac{1}{4} \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{\left(d \cdot d\right) \cdot \ell}\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\left(\frac{1}{4} \cdot D\right) \cdot \color{blue}{\left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)}\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, D \cdot \left(\left(\frac{1}{4} \cdot D\right) \cdot \color{blue}{\left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)}\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

      15. lower-/.f64 32.2%

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot h, D \cdot \left(\left(0.25 \cdot D\right) \cdot \left(M \cdot \color{blue}{\frac{M}{\left(d \cdot d\right) \cdot \ell}}\right)\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 32.2%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot h, \color{blue}{D \cdot \left(\left(0.25 \cdot D\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right)}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}} \]

   if 1.8500000000000001e-303 < l

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot 1 \]

      3. sqrt-prod N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}}\right| \cdot 1 \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}}\right| \cdot 1 \]

      5. lower-unsound-sqrt.f64 N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}}\right| \cdot 1 \]

      6. lower-unsound-sqrt.f64 25.7%

         \[\leadsto \left|\frac{-d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}}\right| \cdot 1 \]

   10. Applied rewrites 25.7%

       \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}}\right| \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+58}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(0.25 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot t\_0\right) \cdot \frac{t\_0}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left(-d\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell} \cdot \sqrt{h}}\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin M (fabs D))) (t_1 (fmax M (fabs D))))
  (if (<= l -2.15e+58)
    (* -1.0 (/ (* d (sqrt (/ -1.0 l))) (sqrt (- h))))
    (if (<= l -5e-310)
      (/
       (*
        (fma
         (* (* (* 0.25 (* t_1 t_1)) t_0) (/ t_0 (* (* d d) l)))
         (* -0.5 h)
         1.0)
        (- d))
       (sqrt (* h l)))
      (* (fabs (/ (- d) (* (sqrt l) (sqrt h)))) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, fabs(D));
	double t_1 = fmax(M, fabs(D));
	double tmp;
	if (l <= -2.15e+58) {
		tmp = -1.0 * ((d * sqrt((-1.0 / l))) / sqrt(-h));
	} else if (l <= -5e-310) {
		tmp = (fma((((0.25 * (t_1 * t_1)) * t_0) * (t_0 / ((d * d) * l))), (-0.5 * h), 1.0) * -d) / sqrt((h * l));
	} else {
		tmp = fabs((-d / (sqrt(l) * sqrt(h)))) * 1.0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fmin(M, abs(D))
	t_1 = fmax(M, abs(D))
	tmp = 0.0
	if (l <= -2.15e+58)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(-1.0 / l))) / sqrt(Float64(-h))));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(0.25 * Float64(t_1 * t_1)) * t_0) * Float64(t_0 / Float64(Float64(d * d) * l))), Float64(-0.5 * h), 1.0) * Float64(-d)) / sqrt(Float64(h * l)));
	else
		tmp = Float64(abs(Float64(Float64(-d) / Float64(sqrt(l) * sqrt(h)))) * 1.0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.15e+58], N[(-1.0 * N[(N[(d * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(N[(N[(N[(N[(0.25 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(t$95$0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + 1.0), $MachinePrecision] * (-d)), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.15e58

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. lower-neg.f64 24.1%

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}} \]

   8. Applied rewrites 24.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}} \]

   if -2.15e58 < l < -4.9999999999999847e-310

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, 1\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot h, \frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \left(-d\right)}{\sqrt{\ell \cdot h}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot h, \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right) \cdot \left(-d\right)}{\sqrt{\ell \cdot h}}} \]

   7. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}, -0.5 \cdot h, 1\right) \cdot \left(-d\right)}{\sqrt{h \cdot \ell}}} \]

   if -4.9999999999999847e-310 < l

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot 1 \]

      3. sqrt-prod N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}}\right| \cdot 1 \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}}\right| \cdot 1 \]

      5. lower-unsound-sqrt.f64 N/A

         \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}}\right| \cdot 1 \]

      6. lower-unsound-sqrt.f64 25.7%

         \[\leadsto \left|\frac{-d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}}\right| \cdot 1 \]

   10. Applied rewrites 25.7%

       \[\leadsto \left|\frac{-d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}}\right| \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\right| \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+234}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right) \cdot 1\\ \end{array} \]

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))))))
  (if (<= t_0 -2e+114)
    (/ (* d (sqrt (/ -1.0 l))) (sqrt (- h)))
    (if (<= t_0 0.0)
      (* (fabs (* d (/ -1.0 (sqrt (* h l))))) 1.0)
      (if (<= t_0 1e+234)
        (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
        (* (* (sqrt (/ 1.0 (* h l))) (fabs d)) 1.0))))))

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 tmp;
	if (t_0 <= -2e+114) {
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	} else if (t_0 <= 0.0) {
		tmp = fabs((d * (-1.0 / sqrt((h * l))))) * 1.0;
	} else if (t_0 <= 1e+234) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (sqrt((1.0 / (h * l))) * fabs(d)) * 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) :: t_0
    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)))
    if (t_0 <= (-2d+114)) then
        tmp = (d * sqrt(((-1.0d0) / l))) / sqrt(-h)
    else if (t_0 <= 0.0d0) then
        tmp = abs((d * ((-1.0d0) / sqrt((h * l))))) * 1.0d0
    else if (t_0 <= 1d+234) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (sqrt((1.0d0 / (h * l))) * abs(d)) * 1.0d0
    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 tmp;
	if (t_0 <= -2e+114) {
		tmp = (d * Math.sqrt((-1.0 / l))) / Math.sqrt(-h);
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d * (-1.0 / Math.sqrt((h * l))))) * 1.0;
	} else if (t_0 <= 1e+234) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (Math.sqrt((1.0 / (h * l))) * Math.abs(d)) * 1.0;
	}
	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)))
	tmp = 0
	if t_0 <= -2e+114:
		tmp = (d * math.sqrt((-1.0 / l))) / math.sqrt(-h)
	elif t_0 <= 0.0:
		tmp = math.fabs((d * (-1.0 / math.sqrt((h * l))))) * 1.0
	elif t_0 <= 1e+234:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (math.sqrt((1.0 / (h * l))) * math.fabs(d)) * 1.0
	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))))
	tmp = 0.0
	if (t_0 <= -2e+114)
		tmp = Float64(Float64(d * sqrt(Float64(-1.0 / l))) / sqrt(Float64(-h)));
	elseif (t_0 <= 0.0)
		tmp = Float64(abs(Float64(d * Float64(-1.0 / sqrt(Float64(h * l))))) * 1.0);
	elseif (t_0 <= 1e+234)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * abs(d)) * 1.0);
	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)));
	tmp = 0.0;
	if (t_0 <= -2e+114)
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	elseif (t_0 <= 0.0)
		tmp = abs((d * (-1.0 / sqrt((h * l))))) * 1.0;
	elseif (t_0 <= 1e+234)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = (sqrt((1.0 / (h * l))) * abs(d)) * 1.0;
	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]}, If[LessEqual[t$95$0, -2e+114], N[(N[(d * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Abs[N[(d * N[(-1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+234], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -2e114

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-neg.f64 3.3%

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}} \]

   8. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}} \]

   if -2e114 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      2. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\left(-d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left|\color{blue}{d} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{d \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left|d \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      8. frac-2neg-rev N/A

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      9. lower-/.f64 43.0%

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

      12. lift-*.f64 43.0%

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

   10. Applied rewrites 43.0%

       \[\leadsto \left|\color{blue}{d \cdot \frac{-1}{\sqrt{h \cdot \ell}}}\right| \cdot 1 \]

   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)))) < 1e234

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. fabs-div N/A

         \[\leadsto \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      5. neg-fabs N/A

         \[\leadsto \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      13. sqrt-undiv N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot 1 \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot 1 \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      17. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot 1 \]

      18. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot 1 \]

      19. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

      21. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      22. lower-unsound-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot 1 \]

      23. lower-/.f64 38.8%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot 1 \]

   10. Applied rewrites 38.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot 1 \]

   if 1e234 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(-d\right)}\right| \cdot 1 \]

      5. fabs-mul N/A

         \[\leadsto \color{blue}{\left(\left|\frac{1}{\sqrt{\ell \cdot h}}\right| \cdot \left|-d\right|\right)} \cdot 1 \]

      6. div-fabs N/A

         \[\leadsto \left(\color{blue}{\frac{\left|1\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot \left|-d\right|\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\left|\sqrt{\ell \cdot h}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|\color{blue}{\mathsf{neg}\left(d\right)}\right|\right) \cdot 1 \]

      12. neg-fabs N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left|d\right|}\right) \cdot 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|d\right|\right)} \cdot 1 \]

   10. Applied rewrites 42.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right)} \cdot 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\right| \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+61}:\\ \;\;\;\;\left|\frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}}\right| \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right) \cdot 1\\ \end{array} \]

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))))))
  (if (<= t_0 -2e+114)
    (/ (* d (sqrt (/ -1.0 l))) (sqrt (- h)))
    (if (<= t_0 0.0)
      (* (fabs (* d (/ -1.0 (sqrt (* h l))))) 1.0)
      (if (<= t_0 1e+61)
        (* (fabs (/ d (* l (sqrt (/ h l))))) 1.0)
        (* (* (sqrt (/ 1.0 (* h l))) (fabs d)) 1.0))))))

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 tmp;
	if (t_0 <= -2e+114) {
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	} else if (t_0 <= 0.0) {
		tmp = fabs((d * (-1.0 / sqrt((h * l))))) * 1.0;
	} else if (t_0 <= 1e+61) {
		tmp = fabs((d / (l * sqrt((h / l))))) * 1.0;
	} else {
		tmp = (sqrt((1.0 / (h * l))) * fabs(d)) * 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) :: t_0
    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)))
    if (t_0 <= (-2d+114)) then
        tmp = (d * sqrt(((-1.0d0) / l))) / sqrt(-h)
    else if (t_0 <= 0.0d0) then
        tmp = abs((d * ((-1.0d0) / sqrt((h * l))))) * 1.0d0
    else if (t_0 <= 1d+61) then
        tmp = abs((d / (l * sqrt((h / l))))) * 1.0d0
    else
        tmp = (sqrt((1.0d0 / (h * l))) * abs(d)) * 1.0d0
    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 tmp;
	if (t_0 <= -2e+114) {
		tmp = (d * Math.sqrt((-1.0 / l))) / Math.sqrt(-h);
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d * (-1.0 / Math.sqrt((h * l))))) * 1.0;
	} else if (t_0 <= 1e+61) {
		tmp = Math.abs((d / (l * Math.sqrt((h / l))))) * 1.0;
	} else {
		tmp = (Math.sqrt((1.0 / (h * l))) * Math.abs(d)) * 1.0;
	}
	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)))
	tmp = 0
	if t_0 <= -2e+114:
		tmp = (d * math.sqrt((-1.0 / l))) / math.sqrt(-h)
	elif t_0 <= 0.0:
		tmp = math.fabs((d * (-1.0 / math.sqrt((h * l))))) * 1.0
	elif t_0 <= 1e+61:
		tmp = math.fabs((d / (l * math.sqrt((h / l))))) * 1.0
	else:
		tmp = (math.sqrt((1.0 / (h * l))) * math.fabs(d)) * 1.0
	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))))
	tmp = 0.0
	if (t_0 <= -2e+114)
		tmp = Float64(Float64(d * sqrt(Float64(-1.0 / l))) / sqrt(Float64(-h)));
	elseif (t_0 <= 0.0)
		tmp = Float64(abs(Float64(d * Float64(-1.0 / sqrt(Float64(h * l))))) * 1.0);
	elseif (t_0 <= 1e+61)
		tmp = Float64(abs(Float64(d / Float64(l * sqrt(Float64(h / l))))) * 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * abs(d)) * 1.0);
	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)));
	tmp = 0.0;
	if (t_0 <= -2e+114)
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	elseif (t_0 <= 0.0)
		tmp = abs((d * (-1.0 / sqrt((h * l))))) * 1.0;
	elseif (t_0 <= 1e+61)
		tmp = abs((d / (l * sqrt((h / l))))) * 1.0;
	else
		tmp = (sqrt((1.0 / (h * l))) * abs(d)) * 1.0;
	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]}, If[LessEqual[t$95$0, -2e+114], N[(N[(d * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Abs[N[(d * N[(-1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+61], N[(N[Abs[N[(d / N[(l * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -2e114

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-neg.f64 3.3%

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}} \]

   8. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}} \]

   if -2e114 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      2. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\left(-d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left|\color{blue}{d} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{d \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left|d \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      8. frac-2neg-rev N/A

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      9. lower-/.f64 43.0%

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

      12. lift-*.f64 43.0%

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

   10. Applied rewrites 43.0%

       \[\leadsto \left|\color{blue}{d \cdot \frac{-1}{\sqrt{h \cdot \ell}}}\right| \cdot 1 \]

   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)))) < 9.9999999999999995e60

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \left|\color{blue}{\frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}}}\right| \cdot 1 \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\color{blue}{\ell \cdot \sqrt{\frac{h}{\ell}}}}\right| \cdot 1 \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\frac{d}{\ell \cdot \color{blue}{\sqrt{\frac{h}{\ell}}}}\right| \cdot 1 \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left|\frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}}\right| \cdot 1 \]

       4. lower-/.f64 39.3%

          \[\leadsto \left|\frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}}\right| \cdot 1 \]

   11. Applied rewrites 39.3%

       \[\leadsto \left|\color{blue}{\frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}}}\right| \cdot 1 \]

   if 9.9999999999999995e60 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(-d\right)}\right| \cdot 1 \]

      5. fabs-mul N/A

         \[\leadsto \color{blue}{\left(\left|\frac{1}{\sqrt{\ell \cdot h}}\right| \cdot \left|-d\right|\right)} \cdot 1 \]

      6. div-fabs N/A

         \[\leadsto \left(\color{blue}{\frac{\left|1\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot \left|-d\right|\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\left|\sqrt{\ell \cdot h}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|\color{blue}{\mathsf{neg}\left(d\right)}\right|\right) \cdot 1 \]

      12. neg-fabs N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left|d\right|}\right) \cdot 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|d\right|\right)} \cdot 1 \]

   10. Applied rewrites 42.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right)} \cdot 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left|d \cdot \frac{-1}{\sqrt{h \cdot \ell}}\right| \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+61}:\\ \;\;\;\;\left|\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\right| \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right) \cdot 1\\ \end{array} \]

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))))))
  (if (<= t_0 -2e+114)
    (/ (* d (sqrt (/ -1.0 l))) (sqrt (- h)))
    (if (<= t_0 0.0)
      (* (fabs (* d (/ -1.0 (sqrt (* h l))))) 1.0)
      (if (<= t_0 1e+61)
        (* (fabs (/ d (* h (sqrt (/ l h))))) 1.0)
        (* (* (sqrt (/ 1.0 (* h l))) (fabs d)) 1.0))))))

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 tmp;
	if (t_0 <= -2e+114) {
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	} else if (t_0 <= 0.0) {
		tmp = fabs((d * (-1.0 / sqrt((h * l))))) * 1.0;
	} else if (t_0 <= 1e+61) {
		tmp = fabs((d / (h * sqrt((l / h))))) * 1.0;
	} else {
		tmp = (sqrt((1.0 / (h * l))) * fabs(d)) * 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) :: t_0
    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)))
    if (t_0 <= (-2d+114)) then
        tmp = (d * sqrt(((-1.0d0) / l))) / sqrt(-h)
    else if (t_0 <= 0.0d0) then
        tmp = abs((d * ((-1.0d0) / sqrt((h * l))))) * 1.0d0
    else if (t_0 <= 1d+61) then
        tmp = abs((d / (h * sqrt((l / h))))) * 1.0d0
    else
        tmp = (sqrt((1.0d0 / (h * l))) * abs(d)) * 1.0d0
    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 tmp;
	if (t_0 <= -2e+114) {
		tmp = (d * Math.sqrt((-1.0 / l))) / Math.sqrt(-h);
	} else if (t_0 <= 0.0) {
		tmp = Math.abs((d * (-1.0 / Math.sqrt((h * l))))) * 1.0;
	} else if (t_0 <= 1e+61) {
		tmp = Math.abs((d / (h * Math.sqrt((l / h))))) * 1.0;
	} else {
		tmp = (Math.sqrt((1.0 / (h * l))) * Math.abs(d)) * 1.0;
	}
	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)))
	tmp = 0
	if t_0 <= -2e+114:
		tmp = (d * math.sqrt((-1.0 / l))) / math.sqrt(-h)
	elif t_0 <= 0.0:
		tmp = math.fabs((d * (-1.0 / math.sqrt((h * l))))) * 1.0
	elif t_0 <= 1e+61:
		tmp = math.fabs((d / (h * math.sqrt((l / h))))) * 1.0
	else:
		tmp = (math.sqrt((1.0 / (h * l))) * math.fabs(d)) * 1.0
	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))))
	tmp = 0.0
	if (t_0 <= -2e+114)
		tmp = Float64(Float64(d * sqrt(Float64(-1.0 / l))) / sqrt(Float64(-h)));
	elseif (t_0 <= 0.0)
		tmp = Float64(abs(Float64(d * Float64(-1.0 / sqrt(Float64(h * l))))) * 1.0);
	elseif (t_0 <= 1e+61)
		tmp = Float64(abs(Float64(d / Float64(h * sqrt(Float64(l / h))))) * 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * abs(d)) * 1.0);
	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)));
	tmp = 0.0;
	if (t_0 <= -2e+114)
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	elseif (t_0 <= 0.0)
		tmp = abs((d * (-1.0 / sqrt((h * l))))) * 1.0;
	elseif (t_0 <= 1e+61)
		tmp = abs((d / (h * sqrt((l / h))))) * 1.0;
	else
		tmp = (sqrt((1.0 / (h * l))) * abs(d)) * 1.0;
	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]}, If[LessEqual[t$95$0, -2e+114], N[(N[(d * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Abs[N[(d * N[(-1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+61], N[(N[Abs[N[(d / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -2e114

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-neg.f64 3.3%

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}} \]

   8. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}} \]

   if -2e114 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      2. frac-2neg N/A

         \[\leadsto \left|\color{blue}{\frac{\mathsf{neg}\left(\left(-d\right)\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\left(-d\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      4. lift-neg.f64 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left|\color{blue}{d} \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{d \cdot \frac{1}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}}\right| \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left|d \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}\right| \cdot 1 \]

      8. frac-2neg-rev N/A

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      9. lower-/.f64 43.0%

         \[\leadsto \left|d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      10. lift-*.f64 N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

      12. lift-*.f64 43.0%

          \[\leadsto \left|d \cdot \frac{-1}{\sqrt{\color{blue}{h \cdot \ell}}}\right| \cdot 1 \]

   10. Applied rewrites 43.0%

       \[\leadsto \left|\color{blue}{d \cdot \frac{-1}{\sqrt{h \cdot \ell}}}\right| \cdot 1 \]

   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)))) < 9.9999999999999995e60

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left|\color{blue}{\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}}\right| \cdot 1 \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{d}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}}\right| \cdot 1 \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\frac{d}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}}\right| \cdot 1 \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left|\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\right| \cdot 1 \]

       4. lower-/.f64 39.2%

          \[\leadsto \left|\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\right| \cdot 1 \]

   11. Applied rewrites 39.2%

       \[\leadsto \left|\color{blue}{\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}}\right| \cdot 1 \]

   if 9.9999999999999995e60 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(-d\right)}\right| \cdot 1 \]

      5. fabs-mul N/A

         \[\leadsto \color{blue}{\left(\left|\frac{1}{\sqrt{\ell \cdot h}}\right| \cdot \left|-d\right|\right)} \cdot 1 \]

      6. div-fabs N/A

         \[\leadsto \left(\color{blue}{\frac{\left|1\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot \left|-d\right|\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\left|\sqrt{\ell \cdot h}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|\color{blue}{\mathsf{neg}\left(d\right)}\right|\right) \cdot 1 \]

      12. neg-fabs N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left|d\right|}\right) \cdot 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|d\right|\right)} \cdot 1 \]

   10. Applied rewrites 42.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right)} \cdot 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right) \cdot 1\\ \end{array} \]

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))))
     -2e+114)
  (/ (* d (sqrt (/ -1.0 l))) (sqrt (- h)))
  (* (* (sqrt (/ 1.0 (* h l))) (fabs d)) 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)))) <= -2e+114) {
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	} else {
		tmp = (sqrt((1.0 / (h * l))) * fabs(d)) * 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)))) <= (-2d+114)) then
        tmp = (d * sqrt(((-1.0d0) / l))) / sqrt(-h)
    else
        tmp = (sqrt((1.0d0 / (h * l))) * abs(d)) * 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)))) <= -2e+114) {
		tmp = (d * Math.sqrt((-1.0 / l))) / Math.sqrt(-h);
	} else {
		tmp = (Math.sqrt((1.0 / (h * l))) * Math.abs(d)) * 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)))) <= -2e+114:
		tmp = (d * math.sqrt((-1.0 / l))) / math.sqrt(-h)
	else:
		tmp = (math.sqrt((1.0 / (h * l))) * math.fabs(d)) * 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)))) <= -2e+114)
		tmp = Float64(Float64(d * sqrt(Float64(-1.0 / l))) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * abs(d)) * 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)))) <= -2e+114)
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	else
		tmp = (sqrt((1.0 / (h * l))) * abs(d)) * 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], -2e+114], N[(N[(d * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[d], $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)))) < -2e114

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-neg.f64 3.3%

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}} \]

   8. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}} \]

   if -2e114 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      3. mult-flip N/A

         \[\leadsto \left|\color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\ell \cdot h}} \cdot \left(-d\right)}\right| \cdot 1 \]

      5. fabs-mul N/A

         \[\leadsto \color{blue}{\left(\left|\frac{1}{\sqrt{\ell \cdot h}}\right| \cdot \left|-d\right|\right)} \cdot 1 \]

      6. div-fabs N/A

         \[\leadsto \left(\color{blue}{\frac{\left|1\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot \left|-d\right|\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\left|\sqrt{\ell \cdot h}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot \left|-d\right|\right) \cdot 1 \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left|-d\right|\right) \cdot 1 \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|\color{blue}{\mathsf{neg}\left(d\right)}\right|\right) \cdot 1 \]

      12. neg-fabs N/A

          \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left|d\right|}\right) \cdot 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot \left|d\right|\right)} \cdot 1 \]

   10. Applied rewrites 42.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left|d\right|\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot 1}{\sqrt{h \cdot \ell}}\\ \end{array} \]

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))))
     -2e+114)
  (/ (* d (sqrt (/ -1.0 l))) (sqrt (- h)))
  (/ (* (fabs d) 1.0) (sqrt (* h l)))))

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)))) <= -2e+114) {
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	} else {
		tmp = (fabs(d) * 1.0) / sqrt((h * l));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: 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)))) <= (-2d+114)) then
        tmp = (d * sqrt(((-1.0d0) / l))) / sqrt(-h)
    else
        tmp = (abs(d) * 1.0d0) / sqrt((h * l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= -2e+114) {
		tmp = (d * Math.sqrt((-1.0 / l))) / Math.sqrt(-h);
	} else {
		tmp = (Math.abs(d) * 1.0) / Math.sqrt((h * l));
	}
	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)))) <= -2e+114:
		tmp = (d * math.sqrt((-1.0 / l))) / math.sqrt(-h)
	else:
		tmp = (math.fabs(d) * 1.0) / math.sqrt((h * l))
	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)))) <= -2e+114)
		tmp = Float64(Float64(d * sqrt(Float64(-1.0 / l))) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(abs(d) * 1.0) / sqrt(Float64(h * l)));
	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)))) <= -2e+114)
		tmp = (d * sqrt((-1.0 / l))) / sqrt(-h);
	else
		tmp = (abs(d) * 1.0) / sqrt((h * l));
	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], -2e+114], N[(N[(d * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] * 1.0), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $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)))) < -2e114

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(\left(\frac{M \cdot D}{d} \cdot 0.25\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(h\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. lower-neg.f64 3.3%

         \[\leadsto \frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}} \]

   8. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{d \cdot \sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}} \]

   if -2e114 < (*.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 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 65.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. sqrt-fabs-rev N/A

         \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lower-fabs.f64 65.2%

          \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   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 43.0%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot 1} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

      4. fabs-div N/A

         \[\leadsto \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot 1 \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      6. neg-fabs N/A

         \[\leadsto \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

      8. sqrt-fabs-rev N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      10. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

   10. Applied rewrites 43.0%

       \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.0% accurate, 7.2× speedup

Math

\[\frac{\left|d\right| \cdot 1}{\sqrt{h \cdot \ell}} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (/ (* (fabs d) 1.0) (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return (fabs(d) * 1.0) / 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 = (abs(d) * 1.0d0) / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.abs(d) * 1.0) / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return (math.fabs(d) * 1.0) / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(Float64(abs(d) * 1.0) / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (abs(d) * 1.0) / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Abs[d], $MachinePrecision] * 1.0), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   18. count-2-rev N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   19. lower-+.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

3. Applied rewrites 65.2%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   4. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   7. metadata-eval N/A

      \[\leadsto {\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   8. pow1/2 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   9. sqrt-fabs-rev N/A

      \[\leadsto \color{blue}{\left|\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right|} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   10. pow1/2 N/A

       \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \left|{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   13. pow-prod-down N/A

       \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   14. lift-pow.f64 N/A

       \[\leadsto \left|\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   15. lift-pow.f64 N/A

       \[\leadsto \left|{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   16. *-commutative N/A

       \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   18. lower-fabs.f64 65.2%

       \[\leadsto \color{blue}{\left|{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \left|\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

5. Applied rewrites 74.7%

   \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

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 43.0%

   \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot 1} \]

   2. lift-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

   3. lift-/.f64 N/A

      \[\leadsto \left|\color{blue}{\frac{-d}{\sqrt{\ell \cdot h}}}\right| \cdot 1 \]

   4. fabs-div N/A

      \[\leadsto \color{blue}{\frac{\left|-d\right|}{\left|\sqrt{\ell \cdot h}\right|}} \cdot 1 \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{\left|\color{blue}{\mathsf{neg}\left(d\right)}\right|}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

   6. neg-fabs N/A

      \[\leadsto \frac{\color{blue}{\left|d\right|}}{\left|\sqrt{\ell \cdot h}\right|} \cdot 1 \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\left|\color{blue}{\sqrt{\ell \cdot h}}\right|} \cdot 1 \]

   8. sqrt-fabs-rev N/A

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

10. Applied rewrites 43.0%

    \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{h \cdot \ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025214 
(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)))))