Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.4% → 86.1%

Time: 11.3s

Alternatives: 19

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.4% 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: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{-0.5}{d} \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{if}\;d \leq -2.35 \cdot 10^{-268}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - t\_2 \cdot \frac{-0.25 \cdot \frac{t\_0 \cdot \left(t\_1 \cdot h\right)}{d}}{\ell}\right)\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-216}:\\ \;\;\;\;-0.125 \cdot \frac{{t\_0}^{2} \cdot {t\_1}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell} \cdot \sqrt{h}}\right| \cdot \left(1 - t\_2 \cdot \frac{\left(\left(\frac{-0.5}{d} \cdot t\_1\right) \cdot t\_0\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmax (fabs M) (fabs D)))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (* (/ -0.5 d) (* t_0 t_1))))
  (if (<= d -2.35e-268)
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (- 1.0 (* t_2 (/ (* -0.25 (/ (* t_0 (* t_1 h)) d)) l))))
    (if (<= d 6e-216)
      (*
       -0.125
       (/ (* (pow t_0 2.0) (pow t_1 2.0)) (* d (* l (sqrt (/ l h))))))
      (*
       (fabs (/ (- d) (* (sqrt l) (sqrt h))))
       (-
        1.0
        (* t_2 (/ (* (* (* (/ -0.5 d) t_1) t_0) (* 0.5 h)) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = (-0.5 / d) * (t_0 * t_1);
	double tmp;
	if (d <= -2.35e-268) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else if (d <= 6e-216) {
		tmp = -0.125 * ((pow(t_0, 2.0) * pow(t_1, 2.0)) / (d * (l * sqrt((l / h)))));
	} else {
		tmp = fabs((-d / (sqrt(l) * sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(m), abs(d_1))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = ((-0.5d0) / d) * (t_0 * t_1)
    if (d <= (-2.35d-268)) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - (t_2 * (((-0.25d0) * ((t_0 * (t_1 * h)) / d)) / l)))
    else if (d <= 6d-216) then
        tmp = (-0.125d0) * (((t_0 ** 2.0d0) * (t_1 ** 2.0d0)) / (d * (l * sqrt((l / h)))))
    else
        tmp = abs((-d / (sqrt(l) * sqrt(h)))) * (1.0d0 - (t_2 * ((((((-0.5d0) / d) * t_1) * t_0) * (0.5d0 * h)) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = (-0.5 / d) * (t_0 * t_1);
	double tmp;
	if (d <= -2.35e-268) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else if (d <= 6e-216) {
		tmp = -0.125 * ((Math.pow(t_0, 2.0) * Math.pow(t_1, 2.0)) / (d * (l * Math.sqrt((l / h)))));
	} else {
		tmp = Math.abs((-d / (Math.sqrt(l) * Math.sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * h)) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = (-0.5 / d) * (t_0 * t_1)
	tmp = 0
	if d <= -2.35e-268:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)))
	elif d <= 6e-216:
		tmp = -0.125 * ((math.pow(t_0, 2.0) * math.pow(t_1, 2.0)) / (d * (l * math.sqrt((l / h)))))
	else:
		tmp = math.fabs((-d / (math.sqrt(l) * math.sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * h)) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(Float64(-0.5 / d) * Float64(t_0 * t_1))
	tmp = 0.0
	if (d <= -2.35e-268)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(t_2 * Float64(Float64(-0.25 * Float64(Float64(t_0 * Float64(t_1 * h)) / d)) / l))));
	elseif (d <= 6e-216)
		tmp = Float64(-0.125 * Float64(Float64((t_0 ^ 2.0) * (t_1 ^ 2.0)) / Float64(d * Float64(l * sqrt(Float64(l / h))))));
	else
		tmp = Float64(abs(Float64(Float64(-d) / Float64(sqrt(l) * sqrt(h)))) * Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(Float64(Float64(-0.5 / d) * t_1) * t_0) * Float64(0.5 * h)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = (-0.5 / d) * (t_0 * t_1);
	tmp = 0.0;
	if (d <= -2.35e-268)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	elseif (d <= 6e-216)
		tmp = -0.125 * (((t_0 ^ 2.0) * (t_1 ^ 2.0)) / (d * (l * sqrt((l / h)))));
	else
		tmp = abs((-d / (sqrt(l) * sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * h)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 / d), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.35e-268], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(N[(-0.25 * N[(N[(t$95$0 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6e-216], N[(-0.125 * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(N[(N[(N[(N[(-0.5 / d), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.3499999999999999e-268

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 75.3%

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

   10. Applied rewrites 75.3%

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

   if -2.3499999999999999e-268 < d < 6.0000000000000003e-216

   1. Initial program 67.4%

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

   3. Applied rewrites 35.9%

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

   5. Applied rewrites 24.3%

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

   6. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{\color{blue}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{\color{blue}{d} \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \color{blue}{\left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \]

      9. lower-/.f64 33.8%

         \[\leadsto -0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \]

   8. Applied rewrites 33.8%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}} \]

   if 6.0000000000000003e-216 < d

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 43.8%

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

   9. Applied rewrites 43.8%

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

Alternative 2: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{-0.5}{d} \cdot \left(D \cdot M\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 10^{-111}:\\ \;\;\;\;\left|\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\right| \cdot \left(1 - t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(M \cdot D\right) \cdot -0.5\right) \cdot \left(0.5 \cdot h\right)}{\ell \cdot d} \cdot \left(\left(D \cdot \frac{-0.5}{d}\right) \cdot M\right)\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

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)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_1 1e-111)
    (*
     (fabs (/ d (* h (sqrt (/ l h)))))
     (- 1.0 (* t_0 (* t_0 (* 0.5 (/ h l))))))
    (if (<= t_1 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (/
       (*
        (-
         1.0
         (*
          (/ (* (* (* M D) -0.5) (* 0.5 h)) (* l d))
          (* (* D (/ -0.5 d)) M)))
        (fabs d))
       (sqrt (* h l)))))))

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))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 1e-111) {
		tmp = fabs((d / (h * sqrt((l / h))))) * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	} else if (t_1 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * fabs(d)) / 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) :: t_0
    real(8) :: t_1
    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))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 1d-111) then
        tmp = abs((d / (h * sqrt((l / h))))) * (1.0d0 - (t_0 * (t_0 * (0.5d0 * (h / l)))))
    else if (t_1 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = ((1.0d0 - (((((m * d_1) * (-0.5d0)) * (0.5d0 * h)) / (l * d)) * ((d_1 * ((-0.5d0) / d)) * m))) * abs(d)) / 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 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))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 1e-111) {
		tmp = Math.abs((d / (h * Math.sqrt((l / h))))) * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	} else if (t_1 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * Math.abs(d)) / Math.sqrt((h * l));
	}
	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))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 1e-111:
		tmp = math.fabs((d / (h * math.sqrt((l / h))))) * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))))
	elif t_1 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * math.fabs(d)) / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(-0.5 / d) * Float64(D * M))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 1e-111)
		tmp = Float64(abs(Float64(d / Float64(h * sqrt(Float64(l / h))))) * Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(0.5 * Float64(h / l))))));
	elseif (t_1 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * D) * -0.5) * Float64(0.5 * h)) / Float64(l * d)) * Float64(Float64(D * Float64(-0.5 / d)) * M))) * abs(d)) / sqrt(Float64(h * l)));
	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))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 1e-111)
		tmp = abs((d / (h * sqrt((l / h))))) * (1.0 - (t_0 * (t_0 * (0.5 * (h / l)))));
	elseif (t_1 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * abs(d)) / sqrt((h * l));
	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[(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$1, 1e-111], N[(N[Abs[N[(d / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 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$1, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(N[(M * D), $MachinePrecision] * -0.5), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $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)))) < 1.0000000000000001e-111

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in h around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 74.7%

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

   8. Applied rewrites 74.7%

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

   if 1.0000000000000001e-111 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

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

   2. 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 38.6%

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

      1. metadata-eval 38.6%

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 38.6%

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

      6. lift-*.f64 N/A

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

   6. Applied rewrites 38.6%

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

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 75.8%

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

Alternative 3: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \frac{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(M \cdot D\right) \cdot -0.5\right) \cdot \left(0.5 \cdot h\right)}{\ell \cdot d} \cdot \left(\left(D \cdot \frac{-0.5}{d}\right) \cdot M\right)\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

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-254)
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (-
      1.0
      (*
       (* (/ -0.5 d) (* D M))
       (/ (* (* (* (/ -0.5 d) M) D) (* 0.5 h)) l))))
    (if (<= t_0 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (/
       (*
        (-
         1.0
         (*
          (/ (* (* (* M D) -0.5) (* 0.5 h)) (* l d))
          (* (* D (/ -0.5 d)) M)))
        (fabs d))
       (sqrt (* h l)))))))

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-254) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (D * M)) * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)));
	} else if (t_0 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * fabs(d)) / 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) :: 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-254) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - ((((-0.5d0) / d) * (d_1 * m)) * ((((((-0.5d0) / d) * m) * d_1) * (0.5d0 * h)) / l)))
    else if (t_0 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = ((1.0d0 - (((((m * d_1) * (-0.5d0)) * (0.5d0 * h)) / (l * d)) * ((d_1 * ((-0.5d0) / d)) * m))) * abs(d)) / 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 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-254) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (D * M)) * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)));
	} else if (t_0 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * Math.abs(d)) / Math.sqrt((h * l));
	}
	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-254:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (D * M)) * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)))
	elif t_0 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * math.fabs(d)) / math.sqrt((h * l))
	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-254)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(-0.5 / d) * Float64(D * M)) * Float64(Float64(Float64(Float64(Float64(-0.5 / d) * M) * D) * Float64(0.5 * h)) / l))));
	elseif (t_0 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * D) * -0.5) * Float64(0.5 * h)) / Float64(l * d)) * Float64(Float64(D * Float64(-0.5 / d)) * M))) * abs(d)) / sqrt(Float64(h * l)));
	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-254)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (D * M)) * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)));
	elseif (t_0 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * ((D * (-0.5 / d)) * M))) * abs(d)) / sqrt((h * l));
	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-254], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(-0.5 / d), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(N[(M * D), $MachinePrecision] * -0.5), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $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)))) < 1.9999999999999998e-254

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

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

   2. 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 38.6%

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

      1. metadata-eval 38.6%

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 38.6%

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

      6. lift-*.f64 N/A

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

   6. Applied rewrites 38.6%

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

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 75.8%

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

Alternative 4: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(D \cdot \frac{-0.5}{d}\right) \cdot M\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - t\_0 \cdot \frac{\left(\left(\frac{-0.5}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(M \cdot D\right) \cdot -0.5\right) \cdot \left(0.5 \cdot h\right)}{\ell \cdot d} \cdot t\_0\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (* D (/ -0.5 d)) M))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_1 2e-254)
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (- 1.0 (* t_0 (/ (* (* (* (/ -0.5 d) M) D) (* 0.5 h)) l))))
    (if (<= t_1 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (/
       (*
        (- 1.0 (* (/ (* (* (* M D) -0.5) (* 0.5 h)) (* l d)) t_0))
        (fabs d))
       (sqrt (* h l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (-0.5 / d)) * M;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 2e-254) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - (t_0 * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)));
	} else if (t_1 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * t_0)) * fabs(d)) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * ((-0.5d0) / d)) * m
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 2d-254) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - (t_0 * ((((((-0.5d0) / d) * m) * d_1) * (0.5d0 * h)) / l)))
    else if (t_1 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = ((1.0d0 - (((((m * d_1) * (-0.5d0)) * (0.5d0 * h)) / (l * d)) * t_0)) * abs(d)) / 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 t_0 = (D * (-0.5 / d)) * M;
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 2e-254) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - (t_0 * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)));
	} else if (t_1 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * t_0)) * Math.abs(d)) / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D * (-0.5 / d)) * M
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 2e-254:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - (t_0 * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)))
	elif t_1 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * t_0)) * math.fabs(d)) / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(-0.5 / d)) * M)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 2e-254)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(t_0 * Float64(Float64(Float64(Float64(Float64(-0.5 / d) * M) * D) * Float64(0.5 * h)) / l))));
	elseif (t_1 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * D) * -0.5) * Float64(0.5 * h)) / Float64(l * d)) * t_0)) * abs(d)) / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (-0.5 / d)) * M;
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 2e-254)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - (t_0 * (((((-0.5 / d) * M) * D) * (0.5 * h)) / l)));
	elseif (t_1 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = ((1.0 - (((((M * D) * -0.5) * (0.5 * h)) / (l * d)) * t_0)) * abs(d)) / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-254], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(N[(N[(N[(-0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(N[(M * D), $MachinePrecision] * -0.5), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $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)))) < 1.9999999999999998e-254

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.2%

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

   9. Applied rewrites 76.2%

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

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

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

   2. 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 38.6%

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

      1. metadata-eval 38.6%

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 38.6%

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

      6. lift-*.f64 N/A

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

   6. Applied rewrites 38.6%

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

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 75.8%

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

Alternative 5: 82.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(\mathsf{max}\left(M, D\right) \cdot \mathsf{min}\left(M, D\right)\right)\right) \cdot \frac{-0.25 \cdot \frac{\mathsf{max}\left(M, D\right) \cdot \left(\mathsf{min}\left(M, D\right) \cdot h\right)}{d}}{\ell}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(t\_0 \cdot -0.5\right) \cdot \left(0.5 \cdot h\right)}{\ell \cdot d} \cdot \left(\left(\mathsf{max}\left(M, D\right) \cdot \frac{-0.5}{d}\right) \cdot \mathsf{min}\left(M, D\right)\right)\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (fmin M D) (fmax M D)))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ t_0 (* 2.0 d)) 2.0)) (/ h l))))))
  (if (<= t_1 2e-254)
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (-
      1.0
      (*
       (* (/ -0.5 d) (* (fmax M D) (fmin M D)))
       (/ (* -0.25 (/ (* (fmax M D) (* (fmin M D) h)) d)) l))))
    (if (<= t_1 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (/
       (*
        (-
         1.0
         (*
          (/ (* (* t_0 -0.5) (* 0.5 h)) (* l d))
          (* (* (fmax M D) (/ -0.5 d)) (fmin M D))))
        (fabs d))
       (sqrt (* h l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, D) * fmax(M, D);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_0 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 2e-254) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (fmax(M, D) * fmin(M, D))) * ((-0.25 * ((fmax(M, D) * (fmin(M, D) * h)) / d)) / l)));
	} else if (t_1 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = ((1.0 - ((((t_0 * -0.5) * (0.5 * h)) / (l * d)) * ((fmax(M, D) * (-0.5 / d)) * fmin(M, D)))) * fabs(d)) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(m, d_1) * fmax(m, d_1)
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_0 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 2d-254) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - ((((-0.5d0) / d) * (fmax(m, d_1) * fmin(m, d_1))) * (((-0.25d0) * ((fmax(m, d_1) * (fmin(m, d_1) * h)) / d)) / l)))
    else if (t_1 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = ((1.0d0 - ((((t_0 * (-0.5d0)) * (0.5d0 * h)) / (l * d)) * ((fmax(m, d_1) * ((-0.5d0) / d)) * fmin(m, d_1)))) * abs(d)) / 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 t_0 = fmin(M, D) * fmax(M, D);
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow((t_0 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 2e-254) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (fmax(M, D) * fmin(M, D))) * ((-0.25 * ((fmax(M, D) * (fmin(M, D) * h)) / d)) / l)));
	} else if (t_1 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = ((1.0 - ((((t_0 * -0.5) * (0.5 * h)) / (l * d)) * ((fmax(M, D) * (-0.5 / d)) * fmin(M, D)))) * Math.abs(d)) / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(M, D) * fmax(M, D)
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow((t_0 / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 2e-254:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (fmax(M, D) * fmin(M, D))) * ((-0.25 * ((fmax(M, D) * (fmin(M, D) * h)) / d)) / l)))
	elif t_1 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = ((1.0 - ((((t_0 * -0.5) * (0.5 * h)) / (l * d)) * ((fmax(M, D) * (-0.5 / d)) * fmin(M, D)))) * math.fabs(d)) / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(fmin(M, D) * fmax(M, D))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_0 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 2e-254)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(-0.5 / d) * Float64(fmax(M, D) * fmin(M, D))) * Float64(Float64(-0.25 * Float64(Float64(fmax(M, D) * Float64(fmin(M, D) * h)) / d)) / l))));
	elseif (t_1 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * -0.5) * Float64(0.5 * h)) / Float64(l * d)) * Float64(Float64(fmax(M, D) * Float64(-0.5 / d)) * fmin(M, D)))) * abs(d)) / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(M, D) * max(M, D);
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((t_0 / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 2e-254)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (max(M, D) * min(M, D))) * ((-0.25 * ((max(M, D) * (min(M, D) * h)) / d)) / l)));
	elseif (t_1 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = ((1.0 - ((((t_0 * -0.5) * (0.5 * h)) / (l * d)) * ((max(M, D) * (-0.5 / d)) * min(M, D)))) * abs(d)) / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$0 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-254], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(-0.5 / d), $MachinePrecision] * N[(N[Max[M, D], $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.25 * N[(N[(N[Max[M, D], $MachinePrecision] * N[(N[Min[M, D], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(t$95$0 * -0.5), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Max[M, D], $MachinePrecision] * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $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)))) < 1.9999999999999998e-254

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 75.3%

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

   10. Applied rewrites 75.3%

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

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

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

   2. 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 38.6%

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

      1. metadata-eval 38.6%

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 38.6%

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

      6. lift-*.f64 N/A

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

   6. Applied rewrites 38.6%

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

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 75.8%

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

Alternative 6: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{-0.5}{d} \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{if}\;\ell \leq 5.45 \cdot 10^{-303}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - t\_2 \cdot \frac{-0.25 \cdot \frac{t\_0 \cdot \left(t\_1 \cdot h\right)}{d}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell} \cdot \sqrt{h}}\right| \cdot \left(1 - t\_2 \cdot \frac{\left(\left(\frac{-0.5}{d} \cdot t\_1\right) \cdot t\_0\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmax (fabs M) (fabs D)))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (* (/ -0.5 d) (* t_0 t_1))))
  (if (<= l 5.45e-303)
    (*
     (fabs (/ (- d) (sqrt (* l h))))
     (- 1.0 (* t_2 (/ (* -0.25 (/ (* t_0 (* t_1 h)) d)) l))))
    (*
     (fabs (/ (- d) (* (sqrt l) (sqrt h))))
     (- 1.0 (* t_2 (/ (* (* (* (/ -0.5 d) t_1) t_0) (* 0.5 h)) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(fabs(M), fabs(D));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = (-0.5 / d) * (t_0 * t_1);
	double tmp;
	if (l <= 5.45e-303) {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else {
		tmp = fabs((-d / (sqrt(l) * sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(m), abs(d_1))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = ((-0.5d0) / d) * (t_0 * t_1)
    if (l <= 5.45d-303) then
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - (t_2 * (((-0.25d0) * ((t_0 * (t_1 * h)) / d)) / l)))
    else
        tmp = abs((-d / (sqrt(l) * sqrt(h)))) * (1.0d0 - (t_2 * ((((((-0.5d0) / d) * t_1) * t_0) * (0.5d0 * h)) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmax(Math.abs(M), Math.abs(D));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = (-0.5 / d) * (t_0 * t_1);
	double tmp;
	if (l <= 5.45e-303) {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	} else {
		tmp = Math.abs((-d / (Math.sqrt(l) * Math.sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * h)) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmax(math.fabs(M), math.fabs(D))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = (-0.5 / d) * (t_0 * t_1)
	tmp = 0
	if l <= 5.45e-303:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)))
	else:
		tmp = math.fabs((-d / (math.sqrt(l) * math.sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * h)) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmax(abs(M), abs(D))
	t_1 = fmin(abs(M), abs(D))
	t_2 = Float64(Float64(-0.5 / d) * Float64(t_0 * t_1))
	tmp = 0.0
	if (l <= 5.45e-303)
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(t_2 * Float64(Float64(-0.25 * Float64(Float64(t_0 * Float64(t_1 * h)) / d)) / l))));
	else
		tmp = Float64(abs(Float64(Float64(-d) / Float64(sqrt(l) * sqrt(h)))) * Float64(1.0 - Float64(t_2 * Float64(Float64(Float64(Float64(Float64(-0.5 / d) * t_1) * t_0) * Float64(0.5 * h)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = max(abs(M), abs(D));
	t_1 = min(abs(M), abs(D));
	t_2 = (-0.5 / d) * (t_0 * t_1);
	tmp = 0.0;
	if (l <= 5.45e-303)
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - (t_2 * ((-0.25 * ((t_0 * (t_1 * h)) / d)) / l)));
	else
		tmp = abs((-d / (sqrt(l) * sqrt(h)))) * (1.0 - (t_2 * (((((-0.5 / d) * t_1) * t_0) * (0.5 * h)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 / d), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5.45e-303], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(N[(-0.25 * N[(N[(t$95$0 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$2 * N[(N[(N[(N[(N[(-0.5 / d), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.45e-303

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 75.3%

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

   10. Applied rewrites 75.3%

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

   if 5.45e-303 < l

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

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

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

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

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

      6. lower-unsound-sqrt.f64 43.8%

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

   9. Applied rewrites 43.8%

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

Alternative 7: 80.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ t_1 := \mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\\ t_2 := 1 - \frac{\left(t\_1 \cdot -0.5\right) \cdot \left(0.5 \cdot h\right)}{\ell \cdot d} \cdot \left(\left(\mathsf{max}\left(M, D\right) \cdot \frac{-0.5}{d}\right) \cdot \mathsf{min}\left(M, D\right)\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot \left|d\right|}{t\_0}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* h l)))
       (t_1 (* (fmin M D) (fmax M D)))
       (t_2
        (-
         1.0
         (*
          (/ (* (* t_1 -0.5) (* 0.5 h)) (* l d))
          (* (* (fmax M D) (/ -0.5 d)) (fmin M D)))))
       (t_3
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ t_1 (* 2.0 d)) 2.0)) (/ h l))))))
  (if (<= t_3 2e-254)
    (* (/ (fabs d) t_0) t_2)
    (if (<= t_3 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (/ (* t_2 (fabs d)) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double t_1 = fmin(M, D) * fmax(M, D);
	double t_2 = 1.0 - ((((t_1 * -0.5) * (0.5 * h)) / (l * d)) * ((fmax(M, D) * (-0.5 / d)) * fmin(M, D)));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_1 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 2e-254) {
		tmp = (fabs(d) / t_0) * t_2;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = (t_2 * fabs(d)) / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((h * l))
    t_1 = fmin(m, d_1) * fmax(m, d_1)
    t_2 = 1.0d0 - ((((t_1 * (-0.5d0)) * (0.5d0 * h)) / (l * d)) * ((fmax(m, d_1) * ((-0.5d0) / d)) * fmin(m, d_1)))
    t_3 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_1 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_3 <= 2d-254) then
        tmp = (abs(d) / t_0) * t_2
    else if (t_3 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = (t_2 * abs(d)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h * l));
	double t_1 = fmin(M, D) * fmax(M, D);
	double t_2 = 1.0 - ((((t_1 * -0.5) * (0.5 * h)) / (l * d)) * ((fmax(M, D) * (-0.5 / d)) * fmin(M, D)));
	double t_3 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow((t_1 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 2e-254) {
		tmp = (Math.abs(d) / t_0) * t_2;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = (t_2 * Math.abs(d)) / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h * l))
	t_1 = fmin(M, D) * fmax(M, D)
	t_2 = 1.0 - ((((t_1 * -0.5) * (0.5 * h)) / (l * d)) * ((fmax(M, D) * (-0.5 / d)) * fmin(M, D)))
	t_3 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow((t_1 / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_3 <= 2e-254:
		tmp = (math.fabs(d) / t_0) * t_2
	elif t_3 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = (t_2 * math.fabs(d)) / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	t_1 = Float64(fmin(M, D) * fmax(M, D))
	t_2 = Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * -0.5) * Float64(0.5 * h)) / Float64(l * d)) * Float64(Float64(fmax(M, D) * Float64(-0.5 / d)) * fmin(M, D))))
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_1 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= 2e-254)
		tmp = Float64(Float64(abs(d) / t_0) * t_2);
	elseif (t_3 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(t_2 * abs(d)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h * l));
	t_1 = min(M, D) * max(M, D);
	t_2 = 1.0 - ((((t_1 * -0.5) * (0.5 * h)) / (l * d)) * ((max(M, D) * (-0.5 / d)) * min(M, D)));
	t_3 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((t_1 / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_3 <= 2e-254)
		tmp = (abs(d) / t_0) * t_2;
	elseif (t_3 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = (t_2 * abs(d)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(N[(t$95$1 * -0.5), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Max[M, D], $MachinePrecision] * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$1 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-254], N[(N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[Abs[d], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999998e-254

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 74.0%

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

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

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

   2. 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 38.6%

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

      1. metadata-eval 38.6%

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 38.6%

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

      6. lift-*.f64 N/A

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

   6. Applied rewrites 38.6%

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

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 75.8%

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

Alternative 8: 80.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(t\_2 \cdot -0.5\right) \cdot \left(0.5 \cdot h\right)}{\ell \cdot d} \cdot \left(\left(t\_1 \cdot \frac{-0.5}{d}\right) \cdot t\_0\right)\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(t\_1 \cdot t\_0\right)\right) \cdot \left(-0.25 \cdot \frac{t\_1 \cdot \left(t\_0 \cdot h\right)}{d \cdot \ell}\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) D))
       (t_1 (fmax (fabs M) D))
       (t_2 (* t_0 t_1))
       (t_3
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ t_2 (* 2.0 d)) 2.0)) (/ h l))))))
  (if (<= t_3 2e-254)
    (*
     (/ (fabs d) (sqrt (* h l)))
     (-
      1.0
      (*
       (/ (* (* t_2 -0.5) (* 0.5 h)) (* l d))
       (* (* t_1 (/ -0.5 d)) t_0))))
    (if (<= t_3 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (*
       (fabs (/ (- d) (sqrt (* l h))))
       (-
        1.0
        (*
         (* (/ -0.5 d) (* t_1 t_0))
         (* -0.25 (/ (* t_1 (* t_0 h)) (* d l))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), D);
	double t_1 = fmax(fabs(M), D);
	double t_2 = t_0 * t_1;
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_2 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 2e-254) {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - ((((t_2 * -0.5) * (0.5 * h)) / (l * d)) * ((t_1 * (-0.5 / d)) * t_0)));
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(m), d_1)
    t_1 = fmax(abs(m), d_1)
    t_2 = t_0 * t_1
    t_3 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * ((t_2 / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_3 <= 2d-254) then
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - ((((t_2 * (-0.5d0)) * (0.5d0 * h)) / (l * d)) * ((t_1 * ((-0.5d0) / d)) * t_0)))
    else if (t_3 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = abs((-d / sqrt((l * h)))) * (1.0d0 - ((((-0.5d0) / d) * (t_1 * t_0)) * ((-0.25d0) * ((t_1 * (t_0 * h)) / (d * l)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(Math.abs(M), D);
	double t_1 = fmax(Math.abs(M), D);
	double t_2 = t_0 * t_1;
	double t_3 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow((t_2 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 2e-254) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - ((((t_2 * -0.5) * (0.5 * h)) / (l * d)) * ((t_1 * (-0.5 / d)) * t_0)));
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), D)
	t_1 = fmax(math.fabs(M), D)
	t_2 = t_0 * t_1
	t_3 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow((t_2 / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_3 <= 2e-254:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - ((((t_2 * -0.5) * (0.5 * h)) / (l * d)) * ((t_1 * (-0.5 / d)) * t_0)))
	elif t_3 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), D)
	t_1 = fmax(abs(M), D)
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_2 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= 2e-254)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_2 * -0.5) * Float64(0.5 * h)) / Float64(l * d)) * Float64(Float64(t_1 * Float64(-0.5 / d)) * t_0))));
	elseif (t_3 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(-0.5 / d) * Float64(t_1 * t_0)) * Float64(-0.25 * Float64(Float64(t_1 * Float64(t_0 * h)) / Float64(d * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), D);
	t_1 = max(abs(M), D);
	t_2 = t_0 * t_1;
	t_3 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((t_2 / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_3 <= 2e-254)
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - ((((t_2 * -0.5) * (0.5 * h)) / (l * d)) * ((t_1 * (-0.5 / d)) * t_0)));
	elseif (t_3 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = abs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$2 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-254], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$2 * -0.5), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(-0.5 / d), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(N[(t$95$1 * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999998e-254

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.8%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 74.0%

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

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

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

   2. 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 38.6%

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

      1. metadata-eval 38.6%

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 38.6%

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

      6. lift-*.f64 N/A

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

   6. Applied rewrites 38.6%

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

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-fabs-rev N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-fabs.f64 53.3%

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

      18. lift-sqrt.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

      22. lift-*.f64 N/A

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

      23. sqrt-div N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 73.6%

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

   8. Applied rewrites 73.6%

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

Alternative 9: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, D\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, D\right)\\ t_2 := \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(t\_1 \cdot t\_0\right)\right) \cdot \left(-0.25 \cdot \frac{t\_1 \cdot \left(t\_0 \cdot h\right)}{d \cdot \ell}\right)\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) D))
       (t_1 (fmax (fabs M) D))
       (t_2
        (*
         (fabs (/ (- d) (sqrt (* l h))))
         (-
          1.0
          (*
           (* (/ -0.5 d) (* t_1 t_0))
           (* -0.25 (/ (* t_1 (* t_0 h)) (* d l)))))))
       (t_3
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_3 2e-254)
    t_2
    (if (<= t_3 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), D);
	double t_1 = fmax(fabs(M), D);
	double t_2 = fabs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 2e-254) {
		tmp = t_2;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(m), d_1)
    t_1 = fmax(abs(m), d_1)
    t_2 = abs((-d / sqrt((l * h)))) * (1.0d0 - ((((-0.5d0) / d) * (t_1 * t_0)) * ((-0.25d0) * ((t_1 * (t_0 * h)) / (d * l)))))
    t_3 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((t_0 * t_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_3 <= 2d-254) then
        tmp = t_2
    else if (t_3 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = 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 = fmin(Math.abs(M), D);
	double t_1 = fmax(Math.abs(M), D);
	double t_2 = Math.abs((-d / Math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))));
	double t_3 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= 2e-254) {
		tmp = t_2;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), D)
	t_1 = fmax(math.fabs(M), D)
	t_2 = math.fabs((-d / math.sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))))
	t_3 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_3 <= 2e-254:
		tmp = t_2
	elif t_3 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), D)
	t_1 = fmax(abs(M), D)
	t_2 = Float64(abs(Float64(Float64(-d) / sqrt(Float64(l * h)))) * Float64(1.0 - Float64(Float64(Float64(-0.5 / d) * Float64(t_1 * t_0)) * Float64(-0.25 * Float64(Float64(t_1 * Float64(t_0 * h)) / Float64(d * l))))))
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= 2e-254)
		tmp = t_2;
	elseif (t_3 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), D);
	t_1 = max(abs(M), D);
	t_2 = abs((-d / sqrt((l * h)))) * (1.0 - (((-0.5 / d) * (t_1 * t_0)) * (-0.25 * ((t_1 * (t_0 * h)) / (d * l)))));
	t_3 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_0 * t_1) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_3 <= 2e-254)
		tmp = t_2;
	elseif (t_3 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], D], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(-0.5 / d), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(N[(t$95$1 * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-254], t$95$2, If[LessEqual[t$95$3, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999998e-254 or 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. lower-fabs.f64 53.3%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{-1}{4} \cdot \color{blue}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{\color{blue}{d} \cdot \ell}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{-1}{4} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)\right) \]

      5. lower-*.f64 73.6%

         \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(-0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\ell}}\right)\right) \]

   8. Applied rewrites 73.6%

      \[\leadsto \left|\frac{-d}{\sqrt{\ell \cdot h}}\right| \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \color{blue}{\left(-0.25 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\right) \]

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

   6. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 76.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ t_1 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_4 := \frac{\left|d\right|}{t\_0}\\ t_5 := t\_4 \cdot 1\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{t\_4 \cdot \left(d - \frac{\left(\left(0.125 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot t\_2\right) \cdot \left(t\_2 \cdot h\right)}{\ell \cdot d}\right)}{d}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(\left(t\_2 \cdot t\_2\right) \cdot t\_1\right) \cdot t\_1\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell} \cdot 0.5\right) \cdot \left|d\right|}{t\_0}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* h l)))
       (t_1 (fmin M (fabs D)))
       (t_2 (fmax M (fabs D)))
       (t_3
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_1 t_2) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_4 (/ (fabs d) t_0))
       (t_5 (* t_4 1.0)))
  (if (<= t_3 -5e-102)
    (/
     (*
      t_4
      (- d (/ (* (* (* 0.125 (* t_1 t_1)) t_2) (* t_2 h)) (* l d))))
     d)
    (if (<= t_3 2e-254)
      t_5
      (if (<= t_3 5e+243)
        (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
        (if (<= t_3 INFINITY)
          t_5
          (/
           (*
            (-
             1.0
             (*
              (/
               (* (* (* (* t_2 t_2) t_1) t_1) h)
               (* (* 4.0 (* d d)) l))
              0.5))
            (fabs d))
           t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double t_1 = fmin(M, fabs(D));
	double t_2 = fmax(M, fabs(D));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double t_4 = fabs(d) / t_0;
	double t_5 = t_4 * 1.0;
	double tmp;
	if (t_3 <= -5e-102) {
		tmp = (t_4 * (d - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / (l * d)))) / d;
	} else if (t_3 <= 2e-254) {
		tmp = t_5;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = ((1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5)) * fabs(d)) / t_0;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h * l));
	double t_1 = fmin(M, Math.abs(D));
	double t_2 = fmax(M, Math.abs(D));
	double t_3 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double t_4 = Math.abs(d) / t_0;
	double t_5 = t_4 * 1.0;
	double tmp;
	if (t_3 <= -5e-102) {
		tmp = (t_4 * (d - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / (l * d)))) / d;
	} else if (t_3 <= 2e-254) {
		tmp = t_5;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_5;
	} else {
		tmp = ((1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5)) * Math.abs(d)) / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h * l))
	t_1 = fmin(M, math.fabs(D))
	t_2 = fmax(M, math.fabs(D))
	t_3 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)))
	t_4 = math.fabs(d) / t_0
	t_5 = t_4 * 1.0
	tmp = 0
	if t_3 <= -5e-102:
		tmp = (t_4 * (d - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / (l * d)))) / d
	elif t_3 <= 2e-254:
		tmp = t_5
	elif t_3 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	elif t_3 <= math.inf:
		tmp = t_5
	else:
		tmp = ((1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5)) * math.fabs(d)) / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	t_1 = fmin(M, abs(D))
	t_2 = fmax(M, abs(D))
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_1 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_4 = Float64(abs(d) / t_0)
	t_5 = Float64(t_4 * 1.0)
	tmp = 0.0
	if (t_3 <= -5e-102)
		tmp = Float64(Float64(t_4 * Float64(d - Float64(Float64(Float64(Float64(0.125 * Float64(t_1 * t_1)) * t_2) * Float64(t_2 * h)) / Float64(l * d)))) / d);
	elseif (t_3 <= 2e-254)
		tmp = t_5;
	elseif (t_3 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_2 * t_2) * t_1) * t_1) * h) / Float64(Float64(4.0 * Float64(d * d)) * l)) * 0.5)) * abs(d)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h * l));
	t_1 = min(M, abs(D));
	t_2 = max(M, abs(D));
	t_3 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_1 * t_2) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_4 = abs(d) / t_0;
	t_5 = t_4 * 1.0;
	tmp = 0.0;
	if (t_3 <= -5e-102)
		tmp = (t_4 * (d - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / (l * d)))) / d;
	elseif (t_3 <= 2e-254)
		tmp = t_5;
	elseif (t_3 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = ((1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5)) * abs(d)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * 1.0), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-102], N[(N[(t$95$4 * N[(d - N[(N[(N[(N[(0.125 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[t$95$3, 2e-254], t$95$5, If[LessEqual[t$95$3, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$5, N[(N[(N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * h), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / t$95$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)))) < -5.0000000000000003e-102

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot 0.5\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}} \cdot \frac{1}{2}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. metadata-eval N/A

         \[\leadsto \left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. associate-*r* N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{\left(\ell \cdot d\right) \cdot d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. associate-/r* N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 42.0%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot 0.5\right)}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   7. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(d - \frac{\left(\left(0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \left(D \cdot h\right)}{\ell \cdot d}\right)}{d}} \]

   if -5.0000000000000003e-102 < (*.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)))) < 1.9999999999999998e-254 or 5.0000000000000004e243 < (*.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 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

   6. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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 69.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. lower-fabs.f64 53.3%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell} \cdot 0.5\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ t_1 := \frac{\left|d\right|}{t\_0}\\ t_2 := t\_1 \cdot 1\\ t_3 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_4 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_5 := 1 - \frac{\left(\left(\left(t\_4 \cdot t\_4\right) \cdot t\_3\right) \cdot t\_3\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell} \cdot 0.5\\ t_6 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3 \cdot t\_4}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{-113}:\\ \;\;\;\;t\_1 \cdot t\_5\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5 \cdot \left|d\right|}{t\_0}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* h l)))
       (t_1 (/ (fabs d) t_0))
       (t_2 (* t_1 1.0))
       (t_3 (fmin M (fabs D)))
       (t_4 (fmax M (fabs D)))
       (t_5
        (-
         1.0
         (*
          (/ (* (* (* (* t_4 t_4) t_3) t_3) h) (* (* 4.0 (* d d)) l))
          0.5)))
       (t_6
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_3 t_4) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_6 -5e-113)
    (* t_1 t_5)
    (if (<= t_6 2e-254)
      t_2
      (if (<= t_6 5e+243)
        (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
        (if (<= t_6 INFINITY) t_2 (/ (* t_5 (fabs d)) t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double t_1 = fabs(d) / t_0;
	double t_2 = t_1 * 1.0;
	double t_3 = fmin(M, fabs(D));
	double t_4 = fmax(M, fabs(D));
	double t_5 = 1.0 - ((((((t_4 * t_4) * t_3) * t_3) * h) / ((4.0 * (d * d)) * l)) * 0.5);
	double t_6 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_3 * t_4) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_6 <= -5e-113) {
		tmp = t_1 * t_5;
	} else if (t_6 <= 2e-254) {
		tmp = t_2;
	} else if (t_6 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (t_5 * fabs(d)) / t_0;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h * l));
	double t_1 = Math.abs(d) / t_0;
	double t_2 = t_1 * 1.0;
	double t_3 = fmin(M, Math.abs(D));
	double t_4 = fmax(M, Math.abs(D));
	double t_5 = 1.0 - ((((((t_4 * t_4) * t_3) * t_3) * h) / ((4.0 * (d * d)) * l)) * 0.5);
	double t_6 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_3 * t_4) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_6 <= -5e-113) {
		tmp = t_1 * t_5;
	} else if (t_6 <= 2e-254) {
		tmp = t_2;
	} else if (t_6 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else if (t_6 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = (t_5 * Math.abs(d)) / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h * l))
	t_1 = math.fabs(d) / t_0
	t_2 = t_1 * 1.0
	t_3 = fmin(M, math.fabs(D))
	t_4 = fmax(M, math.fabs(D))
	t_5 = 1.0 - ((((((t_4 * t_4) * t_3) * t_3) * h) / ((4.0 * (d * d)) * l)) * 0.5)
	t_6 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_3 * t_4) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_6 <= -5e-113:
		tmp = t_1 * t_5
	elif t_6 <= 2e-254:
		tmp = t_2
	elif t_6 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	elif t_6 <= math.inf:
		tmp = t_2
	else:
		tmp = (t_5 * math.fabs(d)) / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	t_1 = Float64(abs(d) / t_0)
	t_2 = Float64(t_1 * 1.0)
	t_3 = fmin(M, abs(D))
	t_4 = fmax(M, abs(D))
	t_5 = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 * t_4) * t_3) * t_3) * h) / Float64(Float64(4.0 * Float64(d * d)) * l)) * 0.5))
	t_6 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_3 * t_4) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_6 <= -5e-113)
		tmp = Float64(t_1 * t_5);
	elseif (t_6 <= 2e-254)
		tmp = t_2;
	elseif (t_6 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	elseif (t_6 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(t_5 * abs(d)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h * l));
	t_1 = abs(d) / t_0;
	t_2 = t_1 * 1.0;
	t_3 = min(M, abs(D));
	t_4 = max(M, abs(D));
	t_5 = 1.0 - ((((((t_4 * t_4) * t_3) * t_3) * h) / ((4.0 * (d * d)) * l)) * 0.5);
	t_6 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_3 * t_4) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_6 <= -5e-113)
		tmp = t_1 * t_5;
	elseif (t_6 <= 2e-254)
		tmp = t_2;
	elseif (t_6 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	elseif (t_6 <= Inf)
		tmp = t_2;
	else
		tmp = (t_5 * abs(d)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision] * h), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$3 * t$95$4), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -5e-113], N[(t$95$1 * t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 2e-254], t$95$2, If[LessEqual[t$95$6, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, Infinity], t$95$2, N[(N[(t$95$5 * N[Abs[d], $MachinePrecision]), $MachinePrecision] / t$95$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)))) < -4.9999999999999997e-113

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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 69.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. lower-fabs.f64 53.3%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell} \cdot 0.5\right)} \]

   if -4.9999999999999997e-113 < (*.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)))) < 1.9999999999999998e-254 or 5.0000000000000004e243 < (*.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 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

   6. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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 69.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. lower-fabs.f64 53.3%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell} \cdot 0.5\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-113}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(\left(t\_2 \cdot t\_2\right) \cdot t\_1\right) \cdot t\_1\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell} \cdot 0.5\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(0.125 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot t\_2\right) \cdot \left(t\_2 \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (fabs d) (sqrt (* h l))))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (fmax (fabs M) (fabs D)))
       (t_3
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_1 t_2) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_3 -5e-113)
    (*
     t_0
     (-
      1.0
      (*
       (/ (* (* (* (* t_2 t_2) t_1) t_1) h) (* (* 4.0 (* d d)) l))
       0.5)))
    (if (<= t_3 2e-254)
      (* t_0 1.0)
      (if (<= t_3 5e+243)
        (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
        (*
         t_0
         (-
          1.0
          (/
           (* (* (* 0.125 (* t_1 t_1)) t_2) (* t_2 h))
           (* (* d d) l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((h * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= -5e-113) {
		tmp = t_0 * (1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5));
	} else if (t_3 <= 2e-254) {
		tmp = t_0 * 1.0;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = abs(d) / sqrt((h * l))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = fmax(abs(m), abs(d_1))
    t_3 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((t_1 * t_2) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_3 <= (-5d-113)) then
        tmp = t_0 * (1.0d0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0d0 * (d * d)) * l)) * 0.5d0))
    else if (t_3 <= 2d-254) then
        tmp = t_0 * 1.0d0
    else if (t_3 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = t_0 * (1.0d0 - ((((0.125d0 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs(d) / Math.sqrt((h * l));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = fmax(Math.abs(M), Math.abs(D));
	double t_3 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= -5e-113) {
		tmp = t_0 * (1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5));
	} else if (t_3 <= 2e-254) {
		tmp = t_0 * 1.0;
	} else if (t_3 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs(d) / math.sqrt((h * l))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = fmax(math.fabs(M), math.fabs(D))
	t_3 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_3 <= -5e-113:
		tmp = t_0 * (1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5))
	elif t_3 <= 2e-254:
		tmp = t_0 * 1.0
	elif t_3 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_1 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= -5e-113)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(t_2 * t_2) * t_1) * t_1) * h) / Float64(Float64(4.0 * Float64(d * d)) * l)) * 0.5)));
	elseif (t_3 <= 2e-254)
		tmp = Float64(t_0 * 1.0);
	elseif (t_3 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(0.125 * Float64(t_1 * t_1)) * t_2) * Float64(t_2 * h)) / Float64(Float64(d * d) * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs(d) / sqrt((h * l));
	t_1 = min(abs(M), abs(D));
	t_2 = max(abs(M), abs(D));
	t_3 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_1 * t_2) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_3 <= -5e-113)
		tmp = t_0 * (1.0 - ((((((t_2 * t_2) * t_1) * t_1) * h) / ((4.0 * (d * d)) * l)) * 0.5));
	elseif (t_3 <= 2e-254)
		tmp = t_0 * 1.0;
	elseif (t_3 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-113], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * h), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-254], N[(t$95$0 * 1.0), $MachinePrecision], If[LessEqual[t$95$3, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(0.125 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999997e-113

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      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 69.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      15. sqrt-fabs-rev N/A

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      17. lower-fabs.f64 53.3%

          \[\leadsto \color{blue}{\left|\sqrt{d \cdot \frac{d}{h \cdot \ell}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \left|\color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      20. lift-/.f64 N/A

          \[\leadsto \left|\sqrt{d \cdot \color{blue}{\frac{d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      23. sqrt-div N/A

          \[\leadsto \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \left(1 - \left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{\frac{-1}{2}}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\left|\frac{-d}{\sqrt{\ell \cdot h}}\right|} \cdot \left(1 - \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell} \cdot 0.5\right)} \]

   if -4.9999999999999997e-113 < (*.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)))) < 1.9999999999999998e-254

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

   6. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot 0.5\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}} \cdot \frac{1}{2}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. metadata-eval N/A

         \[\leadsto \left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. associate-*r* N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{\left(\ell \cdot d\right) \cdot d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. associate-/r* N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 42.0%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot 0.5\right)}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   7. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\left(0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \left(D \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_3 := t\_0 \cdot \left(1 - \frac{\left(\left(0.125 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot t\_2\right) \cdot \left(t\_2 \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}\right)\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_1 \cdot t\_2}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{-102}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (fabs d) (sqrt (* h l))))
       (t_1 (fmin (fabs M) (fabs D)))
       (t_2 (fmax (fabs M) (fabs D)))
       (t_3
        (*
         t_0
         (-
          1.0
          (/
           (* (* (* 0.125 (* t_1 t_1)) t_2) (* t_2 h))
           (* (* d d) l)))))
       (t_4
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_1 t_2) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_4 -5e-102)
    t_3
    (if (<= t_4 2e-254)
      (* t_0 1.0)
      (if (<= t_4 5e+243)
        (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
        t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((h * l));
	double t_1 = fmin(fabs(M), fabs(D));
	double t_2 = fmax(fabs(M), fabs(D));
	double t_3 = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)));
	double t_4 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_4 <= -5e-102) {
		tmp = t_3;
	} else if (t_4 <= 2e-254) {
		tmp = t_0 * 1.0;
	} else if (t_4 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} 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) :: t_4
    real(8) :: tmp
    t_0 = abs(d) / sqrt((h * l))
    t_1 = fmin(abs(m), abs(d_1))
    t_2 = fmax(abs(m), abs(d_1))
    t_3 = t_0 * (1.0d0 - ((((0.125d0 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)))
    t_4 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((t_1 * t_2) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_4 <= (-5d-102)) then
        tmp = t_3
    else if (t_4 <= 2d-254) then
        tmp = t_0 * 1.0d0
    else if (t_4 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    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 = Math.abs(d) / Math.sqrt((h * l));
	double t_1 = fmin(Math.abs(M), Math.abs(D));
	double t_2 = fmax(Math.abs(M), Math.abs(D));
	double t_3 = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)));
	double t_4 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_4 <= -5e-102) {
		tmp = t_3;
	} else if (t_4 <= 2e-254) {
		tmp = t_0 * 1.0;
	} else if (t_4 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs(d) / math.sqrt((h * l))
	t_1 = fmin(math.fabs(M), math.fabs(D))
	t_2 = fmax(math.fabs(M), math.fabs(D))
	t_3 = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)))
	t_4 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_1 * t_2) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_4 <= -5e-102:
		tmp = t_3
	elif t_4 <= 2e-254:
		tmp = t_0 * 1.0
	elif t_4 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = t_3
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_1 = fmin(abs(M), abs(D))
	t_2 = fmax(abs(M), abs(D))
	t_3 = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(0.125 * Float64(t_1 * t_1)) * t_2) * Float64(t_2 * h)) / Float64(Float64(d * d) * l))))
	t_4 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_1 * t_2) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_4 <= -5e-102)
		tmp = t_3;
	elseif (t_4 <= 2e-254)
		tmp = Float64(t_0 * 1.0);
	elseif (t_4 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs(d) / sqrt((h * l));
	t_1 = min(abs(M), abs(D));
	t_2 = max(abs(M), abs(D));
	t_3 = t_0 * (1.0 - ((((0.125 * (t_1 * t_1)) * t_2) * (t_2 * h)) / ((d * d) * l)));
	t_4 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_1 * t_2) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_4 <= -5e-102)
		tmp = t_3;
	elseif (t_4 <= 2e-254)
		tmp = t_0 * 1.0;
	elseif (t_4 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(0.125 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-102], t$95$3, If[LessEqual[t$95$4, 2e-254], N[(t$95$0 * 1.0), $MachinePrecision], If[LessEqual[t$95$4, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $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)))) < -5.0000000000000003e-102 or 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot 0.5\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}} \cdot \frac{1}{2}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      3. metadata-eval N/A

         \[\leadsto \left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      8. associate-*r* N/A

         \[\leadsto \left(1 - \frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{\left(\ell \cdot d\right) \cdot d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      9. associate-/r* N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)\right) \cdot \frac{1}{2}}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   5. Applied rewrites 42.0%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot 0.5\right)}{\ell \cdot d}}{d}}\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}} \]

   7. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\left(0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \left(D \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}\right)} \]

   if -5.0000000000000003e-102 < (*.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)))) < 1.9999999999999998e-254

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

   if 1.9999999999999998e-254 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e243

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

   6. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d - \frac{\left(\left(0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \left(D \cdot h\right)}{\ell \cdot d}}{-t\_1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_1} \cdot 1\\ \end{array} \]

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)))))
       (t_1 (sqrt (* h l))))
  (if (<= t_0 0.0)
    (/ (- d (/ (* (* (* 0.125 (* M M)) D) (* D h)) (* l d))) (- t_1))
    (if (<= t_0 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (* (/ (fabs d) t_1) 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 t_1 = sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d - ((((0.125 * (M * M)) * D) * (D * h)) / (l * d))) / -t_1;
	} else if (t_0 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = (fabs(d) / t_1) * 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) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((h * l))
    if (t_0 <= 0.0d0) then
        tmp = (d - ((((0.125d0 * (m * m)) * d_1) * (d_1 * h)) / (l * d))) / -t_1
    else if (t_0 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = (abs(d) / t_1) * 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 t_1 = Math.sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d - ((((0.125 * (M * M)) * D) * (D * h)) / (l * d))) / -t_1;
	} else if (t_0 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = (Math.abs(d) / t_1) * 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)))
	t_1 = math.sqrt((h * l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (d - ((((0.125 * (M * M)) * D) * (D * h)) / (l * d))) / -t_1
	elif t_0 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = (math.fabs(d) / t_1) * 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))))
	t_1 = sqrt(Float64(h * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(d - Float64(Float64(Float64(Float64(0.125 * Float64(M * M)) * D) * Float64(D * h)) / Float64(l * d))) / Float64(-t_1));
	elseif (t_0 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(abs(d) / t_1) * 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)));
	t_1 = sqrt((h * l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (d - ((((0.125 * (M * M)) * D) * (D * h)) / (l * d))) / -t_1;
	elseif (t_0 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = (abs(d) / t_1) * 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]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(d - N[(N[(N[(N[(0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$0, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\left(1 - \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot 0.5\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]

   5. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot 0.25\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{0.5}{\left(d \cdot d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 34.8%

      \[\leadsto \color{blue}{\frac{d - \frac{\left(\left(0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \left(D \cdot h\right)}{\ell \cdot d}}{-\sqrt{h \cdot \ell}}} \]

   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)))) < 5.0000000000000004e243

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

   6. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 49.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \]

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 0.0)
    (* (* -1.0 (/ (fabs d) (* h (sqrt (/ l h))))) 1.0)
    (if (<= t_0 5e+243)
      (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
      (* (/ (fabs d) (sqrt (* h l))) 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 <= 0.0) {
		tmp = (-1.0 * (fabs(d) / (h * sqrt((l / h))))) * 1.0;
	} else if (t_0 <= 5e+243) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * 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 <= 0.0d0) then
        tmp = ((-1.0d0) * (abs(d) / (h * sqrt((l / h))))) * 1.0d0
    else if (t_0 <= 5d+243) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = (abs(d) / sqrt((h * l))) * 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 <= 0.0) {
		tmp = (-1.0 * (Math.abs(d) / (h * Math.sqrt((l / h))))) * 1.0;
	} else if (t_0 <= 5e+243) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 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 <= 0.0:
		tmp = (-1.0 * (math.fabs(d) / (h * math.sqrt((l / h))))) * 1.0
	elif t_0 <= 5e+243:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 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 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(abs(d) / Float64(h * sqrt(Float64(l / h))))) * 1.0);
	elseif (t_0 <= 5e+243)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 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 <= 0.0)
		tmp = (-1.0 * (abs(d) / (h * sqrt((l / h))))) * 1.0;
	elseif (t_0 <= 5e+243)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = (abs(d) / sqrt((h * l))) * 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, 0.0], N[(N[(-1.0 * N[(N[Abs[d], $MachinePrecision] / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+243], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right)} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}}\right) \cdot 1 \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}}\right) \cdot 1 \]

      3. lower-fabs.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{\color{blue}{h} \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}}\right) \cdot 1 \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1 \]

      6. lower-/.f64 25.4%

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1 \]

   9. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right)} \cdot 1 \]

   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)))) < 5.0000000000000004e243

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 38.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

   6. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 5.0000000000000004e243 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 45.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-113}:\\ \;\;\;\;\left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (if (<=
     (*
      (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
      (-
       1.0
       (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
     -5e-113)
  (* (* -1.0 (/ (fabs d) (* h (sqrt (/ l h))))) 1.0)
  (* (/ (fabs d) (sqrt (* h l))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-113) {
		tmp = (-1.0 * (fabs(d) / (h * sqrt((l / h))))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-113)) then
        tmp = ((-1.0d0) * (abs(d) / (h * sqrt((l / h))))) * 1.0d0
    else
        tmp = (abs(d) / sqrt((h * l))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-113) {
		tmp = (-1.0 * (Math.abs(d) / (h * Math.sqrt((l / h))))) * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-113:
		tmp = (-1.0 * (math.fabs(d) / (h * math.sqrt((l / h))))) * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-113)
		tmp = Float64(Float64(-1.0 * Float64(abs(d) / Float64(h * sqrt(Float64(l / h))))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-113)
		tmp = (-1.0 * (abs(d) / (h * sqrt((l / h))))) * 1.0;
	else
		tmp = (abs(d) / sqrt((h * l))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-113], N[(N[(-1.0 * N[(N[Abs[d], $MachinePrecision] / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999997e-113

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right)} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}}\right) \cdot 1 \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}}\right) \cdot 1 \]

      3. lower-fabs.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{\color{blue}{h} \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}}\right) \cdot 1 \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1 \]

      6. lower-/.f64 25.4%

         \[\leadsto \left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right) \cdot 1 \]

   9. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}\right)} \cdot 1 \]

   if -4.9999999999999997e-113 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (if (<=
     (*
      (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
      (-
       1.0
       (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
     -5e-102)
  (* (/ (fabs d) (* h (sqrt (/ l h)))) 1.0)
  (* (/ (fabs d) (sqrt (* h l))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-102) {
		tmp = (fabs(d) / (h * sqrt((l / h)))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-102)) then
        tmp = (abs(d) / (h * sqrt((l / h)))) * 1.0d0
    else
        tmp = (abs(d) / sqrt((h * l))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-102) {
		tmp = (Math.abs(d) / (h * Math.sqrt((l / h)))) * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-102:
		tmp = (math.fabs(d) / (h * math.sqrt((l / h)))) * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-102)
		tmp = Float64(Float64(abs(d) / Float64(h * sqrt(Float64(l / h)))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-102)
		tmp = (abs(d) / (h * sqrt((l / h)))) * 1.0;
	else
		tmp = (abs(d) / sqrt((h * l))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-102], N[(N[(N[Abs[d], $MachinePrecision] / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000003e-102

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}} \cdot 1 \]

      2. lower-fabs.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{h} \cdot \sqrt{\frac{\ell}{h}}} \cdot 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}} \cdot 1 \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}} \cdot 1 \]

      5. lower-/.f64 26.2%

         \[\leadsto \frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}} \cdot 1 \]

   9. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{h \cdot \sqrt{\frac{\ell}{h}}}} \cdot 1 \]

   if -5.0000000000000003e-102 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. 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 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 41.7% accurate, 12.2× speedup

Math

\[\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1 \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (* (/ (fabs d) (sqrt (* h l))) 1.0))

C

double code(double d, double h, double l, double M, double D) {
	return (fabs(d) / sqrt((h * l))) * 1.0;
}

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) / sqrt((h * l))) * 1.0d0
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
}

Python

def code(d, h, l, M, D):
	return (math.fabs(d) / math.sqrt((h * l))) * 1.0

Julia

function code(d, h, l, M, D)
	return Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (abs(d) / sqrt((h * l))) * 1.0;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 67.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. 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 38.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
5. Step-by-step derivation

6. Applied rewrites 41.7%

   \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]
7. Add Preprocessing

Alternative 19: 41.7% accurate, 12.2× speedup

Math

\[\left|d\right| \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

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(abs(d) * Float64(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[Abs[d], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. 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 38.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
5. Step-by-step derivation

6. Applied rewrites 41.7%

   \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]

8. Applied rewrites 41.7%

   \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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)))))