Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 79.2%

Time: 8.0s

Alternatives: 14

Speedup: 3.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := 1 - \frac{\frac{t\_0 \cdot t\_0}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot \sqrt{d}}{\sqrt{h}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m))
        (t_1 (- 1.0 (/ (* (/ (* t_0 t_0) 4.0) (* 0.5 h)) l)))
        (t_2 (sqrt (/ d l))))
   (if (<= d -5e-310)
     (* (* t_2 (/ (sqrt (- d)) (sqrt (- h)))) t_1)
     (if (<= d 2.1e-101)
       (* (/ (* (sqrt d) (sqrt d)) (sqrt (* h l))) t_1)
       (* (/ (* t_2 (sqrt d)) (sqrt h)) t_1)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l);
	double t_2 = sqrt((d / l));
	double tmp;
	if (d <= -5e-310) {
		tmp = (t_2 * (sqrt(-d) / sqrt(-h))) * t_1;
	} else if (d <= 2.1e-101) {
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_1;
	} else {
		tmp = ((t_2 * sqrt(d)) / sqrt(h)) * t_1;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (m_m / d) * d_m
    t_1 = 1.0d0 - ((((t_0 * t_0) / 4.0d0) * (0.5d0 * h)) / l)
    t_2 = sqrt((d / l))
    if (d <= (-5d-310)) then
        tmp = (t_2 * (sqrt(-d) / sqrt(-h))) * t_1
    else if (d <= 2.1d-101) then
        tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_1
    else
        tmp = ((t_2 * sqrt(d)) / sqrt(h)) * t_1
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l);
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (d <= -5e-310) {
		tmp = (t_2 * (Math.sqrt(-d) / Math.sqrt(-h))) * t_1;
	} else if (d <= 2.1e-101) {
		tmp = ((Math.sqrt(d) * Math.sqrt(d)) / Math.sqrt((h * l))) * t_1;
	} else {
		tmp = ((t_2 * Math.sqrt(d)) / Math.sqrt(h)) * t_1;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (M_m / d) * D_m
	t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l)
	t_2 = math.sqrt((d / l))
	tmp = 0
	if d <= -5e-310:
		tmp = (t_2 * (math.sqrt(-d) / math.sqrt(-h))) * t_1
	elif d <= 2.1e-101:
		tmp = ((math.sqrt(d) * math.sqrt(d)) / math.sqrt((h * l))) * t_1
	else:
		tmp = ((t_2 * math.sqrt(d)) / math.sqrt(h)) * t_1
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(0.5 * h)) / l))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(t_2 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * t_1);
	elseif (d <= 2.1e-101)
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(d)) / sqrt(Float64(h * l))) * t_1);
	else
		tmp = Float64(Float64(Float64(t_2 * sqrt(d)) / sqrt(h)) * t_1);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (M_m / d) * D_m;
	t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l);
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = (t_2 * (sqrt(-d) / sqrt(-h))) * t_1;
	elseif (d <= 2.1e-101)
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_1;
	else
		tmp = ((t_2 * sqrt(d)) / sqrt(h)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(t$95$2 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 2.1e-101], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(t$95$2 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 66.8

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

   3. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 68.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 68.2

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 68.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 78.7

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 78.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if -4.999999999999985e-310 < d < 2.10000000000000016e-101

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 48.5

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

   3. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 49.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 49.1

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 49.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. frac-times N/A

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

      9. sqrt-prod N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 65.3

          \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 2.10000000000000016e-101 < d

   1. Initial program 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 75.6

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

   3. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 78.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 78.1

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 86.6

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 86.6%

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

Alternative 2: 73.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{M\_m}{d} \cdot D\_m\\ t_2 := 1 - \frac{\frac{t\_1 \cdot t\_1}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\\ \mathbf{if}\;h \leq -3.3 \cdot 10^{+227}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}} \cdot t\_2\\ \mathbf{elif}\;h \leq 3.85 \cdot 10^{-295}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_2\\ \mathbf{elif}\;h \leq 7.9 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \sqrt{d}}{\sqrt{h}} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (* (/ M_m d) D_m))
        (t_2 (- 1.0 (/ (* (/ (* t_1 t_1) 4.0) (* 0.5 h)) l))))
   (if (<= h -3.3e+227)
     (* (/ (sqrt (* (/ d l) (- d))) (sqrt (- h))) t_2)
     (if (<= h 3.85e-295)
       (* (* t_0 (sqrt (/ d h))) t_2)
       (if (<= h 7.9e-41)
         (* (/ (* (sqrt d) (sqrt d)) (sqrt (* h l))) t_2)
         (* (/ (* t_0 (sqrt d)) (sqrt h)) t_2))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (M_m / d) * D_m;
	double t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l);
	double tmp;
	if (h <= -3.3e+227) {
		tmp = (sqrt(((d / l) * -d)) / sqrt(-h)) * t_2;
	} else if (h <= 3.85e-295) {
		tmp = (t_0 * sqrt((d / h))) * t_2;
	} else if (h <= 7.9e-41) {
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_2;
	} else {
		tmp = ((t_0 * sqrt(d)) / sqrt(h)) * t_2;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (m_m / d) * d_m
    t_2 = 1.0d0 - ((((t_1 * t_1) / 4.0d0) * (0.5d0 * h)) / l)
    if (h <= (-3.3d+227)) then
        tmp = (sqrt(((d / l) * -d)) / sqrt(-h)) * t_2
    else if (h <= 3.85d-295) then
        tmp = (t_0 * sqrt((d / h))) * t_2
    else if (h <= 7.9d-41) then
        tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_2
    else
        tmp = ((t_0 * sqrt(d)) / sqrt(h)) * t_2
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (M_m / d) * D_m;
	double t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l);
	double tmp;
	if (h <= -3.3e+227) {
		tmp = (Math.sqrt(((d / l) * -d)) / Math.sqrt(-h)) * t_2;
	} else if (h <= 3.85e-295) {
		tmp = (t_0 * Math.sqrt((d / h))) * t_2;
	} else if (h <= 7.9e-41) {
		tmp = ((Math.sqrt(d) * Math.sqrt(d)) / Math.sqrt((h * l))) * t_2;
	} else {
		tmp = ((t_0 * Math.sqrt(d)) / Math.sqrt(h)) * t_2;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (M_m / d) * D_m
	t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l)
	tmp = 0
	if h <= -3.3e+227:
		tmp = (math.sqrt(((d / l) * -d)) / math.sqrt(-h)) * t_2
	elif h <= 3.85e-295:
		tmp = (t_0 * math.sqrt((d / h))) * t_2
	elif h <= 7.9e-41:
		tmp = ((math.sqrt(d) * math.sqrt(d)) / math.sqrt((h * l))) * t_2
	else:
		tmp = ((t_0 * math.sqrt(d)) / math.sqrt(h)) * t_2
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(M_m / d) * D_m)
	t_2 = Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) / 4.0) * Float64(0.5 * h)) / l))
	tmp = 0.0
	if (h <= -3.3e+227)
		tmp = Float64(Float64(sqrt(Float64(Float64(d / l) * Float64(-d))) / sqrt(Float64(-h))) * t_2);
	elseif (h <= 3.85e-295)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * t_2);
	elseif (h <= 7.9e-41)
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(d)) / sqrt(Float64(h * l))) * t_2);
	else
		tmp = Float64(Float64(Float64(t_0 * sqrt(d)) / sqrt(h)) * t_2);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = (M_m / d) * D_m;
	t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l);
	tmp = 0.0;
	if (h <= -3.3e+227)
		tmp = (sqrt(((d / l) * -d)) / sqrt(-h)) * t_2;
	elseif (h <= 3.85e-295)
		tmp = (t_0 * sqrt((d / h))) * t_2;
	elseif (h <= 7.9e-41)
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_2;
	else
		tmp = ((t_0 * sqrt(d)) / sqrt(h)) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.3e+227], N[(N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * (-d)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, 3.85e-295], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, 7.9e-41], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -3.2999999999999999e227

   1. Initial program 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 41.6

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

   3. Applied rewrites 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 43.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 43.2

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 43.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 52.5

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell} \cdot \left(-d\right)}}}{\sqrt{-h}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if -3.2999999999999999e227 < h < 3.85000000000000009e-295

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 70.0

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

   3. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 71.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 71.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 71.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 3.85000000000000009e-295 < h < 7.8999999999999999e-41

   1. Initial program 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 70.3

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

   3. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 71.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 71.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. frac-times N/A

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

      9. sqrt-prod N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 82.6

          \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 7.8999999999999999e-41 < h

   1. Initial program 64.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 64.4

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

   3. Applied rewrites 64.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 66.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 66.6

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 75.3

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 75.3%

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

Alternative 3: 73.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{M\_m}{d} \cdot D\_m\\ t_2 := 1 - \frac{\frac{t\_1 \cdot t\_1}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\\ \mathbf{if}\;h \leq -3.3 \cdot 10^{+227}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}} \cdot t\_2\\ \mathbf{elif}\;h \leq 3.85 \cdot 10^{-295}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_0\right) \cdot t\_2\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \sqrt{d}}{\sqrt{\ell}} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (* (/ M_m d) D_m))
        (t_2 (- 1.0 (/ (* (/ (* t_1 t_1) 4.0) (* 0.5 h)) l))))
   (if (<= h -3.3e+227)
     (* (/ (sqrt (* (/ d l) (- d))) (sqrt (- h))) t_2)
     (if (<= h 3.85e-295)
       (* (* (sqrt (/ d l)) t_0) t_2)
       (if (<= h 3.5e-95)
         (* (/ (* (sqrt d) (sqrt d)) (sqrt (* h l))) t_2)
         (* (/ (* t_0 (sqrt d)) (sqrt l)) t_2))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = (M_m / d) * D_m;
	double t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l);
	double tmp;
	if (h <= -3.3e+227) {
		tmp = (sqrt(((d / l) * -d)) / sqrt(-h)) * t_2;
	} else if (h <= 3.85e-295) {
		tmp = (sqrt((d / l)) * t_0) * t_2;
	} else if (h <= 3.5e-95) {
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_2;
	} else {
		tmp = ((t_0 * sqrt(d)) / sqrt(l)) * t_2;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = (m_m / d) * d_m
    t_2 = 1.0d0 - ((((t_1 * t_1) / 4.0d0) * (0.5d0 * h)) / l)
    if (h <= (-3.3d+227)) then
        tmp = (sqrt(((d / l) * -d)) / sqrt(-h)) * t_2
    else if (h <= 3.85d-295) then
        tmp = (sqrt((d / l)) * t_0) * t_2
    else if (h <= 3.5d-95) then
        tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_2
    else
        tmp = ((t_0 * sqrt(d)) / sqrt(l)) * t_2
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = (M_m / d) * D_m;
	double t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l);
	double tmp;
	if (h <= -3.3e+227) {
		tmp = (Math.sqrt(((d / l) * -d)) / Math.sqrt(-h)) * t_2;
	} else if (h <= 3.85e-295) {
		tmp = (Math.sqrt((d / l)) * t_0) * t_2;
	} else if (h <= 3.5e-95) {
		tmp = ((Math.sqrt(d) * Math.sqrt(d)) / Math.sqrt((h * l))) * t_2;
	} else {
		tmp = ((t_0 * Math.sqrt(d)) / Math.sqrt(l)) * t_2;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / h))
	t_1 = (M_m / d) * D_m
	t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l)
	tmp = 0
	if h <= -3.3e+227:
		tmp = (math.sqrt(((d / l) * -d)) / math.sqrt(-h)) * t_2
	elif h <= 3.85e-295:
		tmp = (math.sqrt((d / l)) * t_0) * t_2
	elif h <= 3.5e-95:
		tmp = ((math.sqrt(d) * math.sqrt(d)) / math.sqrt((h * l))) * t_2
	else:
		tmp = ((t_0 * math.sqrt(d)) / math.sqrt(l)) * t_2
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64(M_m / d) * D_m)
	t_2 = Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) / 4.0) * Float64(0.5 * h)) / l))
	tmp = 0.0
	if (h <= -3.3e+227)
		tmp = Float64(Float64(sqrt(Float64(Float64(d / l) * Float64(-d))) / sqrt(Float64(-h))) * t_2);
	elseif (h <= 3.85e-295)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_0) * t_2);
	elseif (h <= 3.5e-95)
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(d)) / sqrt(Float64(h * l))) * t_2);
	else
		tmp = Float64(Float64(Float64(t_0 * sqrt(d)) / sqrt(l)) * t_2);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / h));
	t_1 = (M_m / d) * D_m;
	t_2 = 1.0 - ((((t_1 * t_1) / 4.0) * (0.5 * h)) / l);
	tmp = 0.0;
	if (h <= -3.3e+227)
		tmp = (sqrt(((d / l) * -d)) / sqrt(-h)) * t_2;
	elseif (h <= 3.85e-295)
		tmp = (sqrt((d / l)) * t_0) * t_2;
	elseif (h <= 3.5e-95)
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_2;
	else
		tmp = ((t_0 * sqrt(d)) / sqrt(l)) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.3e+227], N[(N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * (-d)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, 3.85e-295], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, 3.5e-95], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -3.2999999999999999e227

   1. Initial program 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 41.6

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

   3. Applied rewrites 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 43.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 43.2

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 43.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 52.5

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell} \cdot \left(-d\right)}}}{\sqrt{-h}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(-d\right)}}{\sqrt{-h}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if -3.2999999999999999e227 < h < 3.85000000000000009e-295

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 70.0

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

   3. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 71.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 71.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 71.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 3.85000000000000009e-295 < h < 3.4999999999999997e-95

   1. Initial program 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 68.9

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

   3. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 70.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 70.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. frac-times N/A

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

      9. sqrt-prod N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 81.1

          \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 3.4999999999999997e-95 < h

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 66.0

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

   3. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 67.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 67.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 75.2

          \[\leadsto \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 75.2%

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

Alternative 4: 72.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := 1 - \frac{\frac{t\_0 \cdot t\_0}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\\ \mathbf{if}\;\ell \leq 1.38 \cdot 10^{-229}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m))
        (t_1 (- 1.0 (/ (* (/ (* t_0 t_0) 4.0) (* 0.5 h)) l))))
   (if (<= l 1.38e-229)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)
     (if (<= l 2e+141)
       (* (/ (* (sqrt d) (sqrt d)) (sqrt (* h l))) t_1)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l);
	double tmp;
	if (l <= 1.38e-229) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1;
	} else if (l <= 2e+141) {
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_1;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m / d) * d_m
    t_1 = 1.0d0 - ((((t_0 * t_0) / 4.0d0) * (0.5d0 * h)) / l)
    if (l <= 1.38d-229) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1
    else if (l <= 2d+141) then
        tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_1
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l);
	double tmp;
	if (l <= 1.38e-229) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * t_1;
	} else if (l <= 2e+141) {
		tmp = ((Math.sqrt(d) * Math.sqrt(d)) / Math.sqrt((h * l))) * t_1;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (M_m / d) * D_m
	t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l)
	tmp = 0
	if l <= 1.38e-229:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * t_1
	elif l <= 2e+141:
		tmp = ((math.sqrt(d) * math.sqrt(d)) / math.sqrt((h * l))) * t_1
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(0.5 * h)) / l))
	tmp = 0.0
	if (l <= 1.38e-229)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_1);
	elseif (l <= 2e+141)
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(d)) / sqrt(Float64(h * l))) * t_1);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (M_m / d) * D_m;
	t_1 = 1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l);
	tmp = 0.0;
	if (l <= 1.38e-229)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1;
	elseif (l <= 2e+141)
		tmp = ((sqrt(d) * sqrt(d)) / sqrt((h * l))) * t_1;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.38e-229], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 2e+141], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.38e-229

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 67.3

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

   3. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 69.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 69.1

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 69.1%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 1.38e-229 < l < 2.00000000000000003e141

   1. Initial program 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 72.1

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

   3. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 74.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 74.6

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. frac-times N/A

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

      9. sqrt-prod N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 82.4

          \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 2.00000000000000003e141 < l

   1. Initial program 51.9%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 50.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 50.2

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

   6. Applied rewrites 50.2%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 65.7

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

   8. Applied rewrites 65.7%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq 4.9 \cdot 10^{+79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{t\_0 \cdot t\_0}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l 4.9e+79)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (/ (* t_0 t_0) 4.0) (* 0.5 h)) l)))
     (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= 4.9e+79) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m / d) * d_m
    if (l <= 4.9d+79) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((t_0 * t_0) / 4.0d0) * (0.5d0 * h)) / l))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= 4.9e+79) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (M_m / d) * D_m
	tmp = 0
	if l <= 4.9e+79:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= 4.9e+79)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(0.5 * h)) / l)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (M_m / d) * D_m;
	tmp = 0.0;
	if (l <= 4.9e+79)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, 4.9e+79], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.8999999999999999e79

   1. Initial program 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 69.1

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

   3. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 71.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 71.2

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 71.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 4.8999999999999999e79 < l

   1. Initial program 56.0%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 49.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 62.8

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

   8. Applied rewrites 62.8%

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

Alternative 6: 64.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{-135}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{M\_m \cdot t\_0}{4} \cdot D\_m\right) \cdot \left(0.5 \cdot h\right)}{d \cdot \ell}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\frac{t\_0 \cdot t\_0}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l -5.4e-135)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* (/ (* M_m t_0) 4.0) D_m) (* 0.5 h)) (* d l))))
     (if (<= l 1.45e+79)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (- 1.0 (/ (* (/ (* t_0 t_0) 4.0) (* 0.5 h)) l)))
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -5.4e-135) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((M_m * t_0) / 4.0) * D_m) * (0.5 * h)) / (d * l)));
	} else if (l <= 1.45e+79) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m / d) * d_m
    if (l <= (-5.4d-135)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((((m_m * t_0) / 4.0d0) * d_m) * (0.5d0 * h)) / (d * l)))
    else if (l <= 1.45d+79) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - ((((t_0 * t_0) / 4.0d0) * (0.5d0 * h)) / l))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -5.4e-135) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((((M_m * t_0) / 4.0) * D_m) * (0.5 * h)) / (d * l)));
	} else if (l <= 1.45e+79) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (M_m / d) * D_m
	tmp = 0
	if l <= -5.4e-135:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((((M_m * t_0) / 4.0) * D_m) * (0.5 * h)) / (d * l)))
	elif l <= 1.45e+79:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= -5.4e-135)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * t_0) / 4.0) * D_m) * Float64(0.5 * h)) / Float64(d * l))));
	elseif (l <= 1.45e+79)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(0.5 * h)) / l)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (M_m / d) * D_m;
	tmp = 0.0;
	if (l <= -5.4e-135)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((M_m * t_0) / 4.0) * D_m) * (0.5 * h)) / (d * l)));
	elseif (l <= 1.45e+79)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -5.4e-135], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(M$95$m * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+79], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.39999999999999997e-135

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 65.1

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

   3. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

   7. Applied rewrites 62.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{M \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot D\right) \cdot \left(0.5 \cdot h\right)}{d \cdot \ell}}\right) \]

   if -5.39999999999999997e-135 < l < 1.44999999999999996e79

   1. Initial program 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 72.2

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

   3. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 75.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 75.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 66.0

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 1.44999999999999996e79 < l

   1. Initial program 56.0%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 49.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      7. lower-sqrt.f64 62.7

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 62.7%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq -1.26 \cdot 10^{-83}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\frac{t\_0 \cdot t\_0}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l -1.26e-83)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (-
       1.0
       (/
        (* (/ (* (* M_m D_m) (* M_m D_m)) (* (* 2.0 d) (* 2.0 d))) (* 0.5 h))
        l)))
     (if (<= l 1.45e+79)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (- 1.0 (/ (* (/ (* t_0 t_0) 4.0) (* 0.5 h)) l)))
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -1.26e-83) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((M_m * D_m) * (M_m * D_m)) / ((2.0 * d) * (2.0 * d))) * (0.5 * h)) / l));
	} else if (l <= 1.45e+79) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m / d) * d_m
    if (l <= (-1.26d-83)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((((m_m * d_m) * (m_m * d_m)) / ((2.0d0 * d) * (2.0d0 * d))) * (0.5d0 * h)) / l))
    else if (l <= 1.45d+79) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - ((((t_0 * t_0) / 4.0d0) * (0.5d0 * h)) / l))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -1.26e-83) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((((M_m * D_m) * (M_m * D_m)) / ((2.0 * d) * (2.0 * d))) * (0.5 * h)) / l));
	} else if (l <= 1.45e+79) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (M_m / d) * D_m
	tmp = 0
	if l <= -1.26e-83:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((((M_m * D_m) * (M_m * D_m)) / ((2.0 * d) * (2.0 * d))) * (0.5 * h)) / l))
	elif l <= 1.45e+79:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= -1.26e-83)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(Float64(2.0 * d) * Float64(2.0 * d))) * Float64(0.5 * h)) / l)));
	elseif (l <= 1.45e+79)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(0.5 * h)) / l)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (M_m / d) * D_m;
	tmp = 0.0;
	if (l <= -1.26e-83)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((M_m * D_m) * (M_m * D_m)) / ((2.0 * d) * (2.0 * d))) * (0.5 * h)) / l));
	elseif (l <= 1.45e+79)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -1.26e-83], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+79], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.2600000000000001e-83

   1. Initial program 63.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 63.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   5. Applied rewrites 63.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(2 \cdot d\right)} \cdot \left(2 \cdot d\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 51.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \color{blue}{\left(2 \cdot d\right)}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 51.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if -1.2600000000000001e-83 < l < 1.44999999999999996e79

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 72.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   5. Applied rewrites 76.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 76.0

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 76.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. lower-*.f64 66.0

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 1.44999999999999996e79 < l

   1. Initial program 56.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 49.8

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      7. lower-sqrt.f64 49.7

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      10. lower-*.f64 49.7

          \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   6. Applied rewrites 49.7%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      7. lower-sqrt.f64 62.7

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 62.7%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+102}:\\ \;\;\;\;-\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\frac{t\_0 \cdot t\_0}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l -7.8e+102)
     (- (/ d (sqrt (* h l))))
     (if (<= l 1.45e+79)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (- 1.0 (/ (* (/ (* t_0 t_0) 4.0) (* 0.5 h)) l)))
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -7.8e+102) {
		tmp = -(d / sqrt((h * l)));
	} else if (l <= 1.45e+79) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m / d) * d_m
    if (l <= (-7.8d+102)) then
        tmp = -(d / sqrt((h * l)))
    else if (l <= 1.45d+79) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - ((((t_0 * t_0) / 4.0d0) * (0.5d0 * h)) / l))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -7.8e+102) {
		tmp = -(d / Math.sqrt((h * l)));
	} else if (l <= 1.45e+79) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (M_m / d) * D_m
	tmp = 0
	if l <= -7.8e+102:
		tmp = -(d / math.sqrt((h * l)))
	elif l <= 1.45e+79:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= -7.8e+102)
		tmp = Float64(-Float64(d / sqrt(Float64(h * l))));
	elseif (l <= 1.45e+79)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(0.5 * h)) / l)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (M_m / d) * D_m;
	tmp = 0.0;
	if (l <= -7.8e+102)
		tmp = -(d / sqrt((h * l)));
	elseif (l <= 1.45e+79)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - ((((t_0 * t_0) / 4.0) * (0.5 * h)) / l));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -7.8e+102], (-N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), If[LessEqual[l, 1.45e+79], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.7999999999999997e102

   1. Initial program 55.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 50.2

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      7. lift-pow.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      8. unpow-1 N/A

         \[\leadsto -d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      9. sqrt-div N/A

         \[\leadsto -d \cdot \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \]

      10. metadata-eval N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      11. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. *-commutative N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      15. associate-*r/ N/A

          \[\leadsto -\frac{d \cdot 1}{\sqrt{h \cdot \ell}} \]

      16. *-commutative N/A

          \[\leadsto -\frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      17. *-lft-identity N/A

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

      18. lower-/.f64 50.2

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 50.2%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -7.7999999999999997e102 < l < 1.44999999999999996e79

   1. Initial program 72.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 72.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 75.5

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 75.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. lower-*.f64 63.8

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 1.44999999999999996e79 < l

   1. Initial program 56.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 49.8

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      7. lower-sqrt.f64 49.7

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      10. lower-*.f64 49.7

          \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   6. Applied rewrites 49.7%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      7. lower-sqrt.f64 62.7

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 62.7%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot \frac{M\_m}{d}\right)}{d \cdot 4} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 4.5e+79)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (-
     1.0
     (/ (* (/ (* (* D_m M_m) (* D_m (/ M_m d))) (* d 4.0)) (* 0.5 h)) l)))
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 4.5e+79) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((D_m * M_m) * (D_m * (M_m / d))) / (d * 4.0)) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 4.5d+79) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((((d_m * m_m) * (d_m * (m_m / d))) / (d * 4.0d0)) * (0.5d0 * h)) / l))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 4.5e+79) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((((D_m * M_m) * (D_m * (M_m / d))) / (d * 4.0)) * (0.5 * h)) / l));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 4.5e+79:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((((D_m * M_m) * (D_m * (M_m / d))) / (d * 4.0)) * (0.5 * h)) / l))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 4.5e+79)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * Float64(M_m / d))) / Float64(d * 4.0)) * Float64(0.5 * h)) / l)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 4.5e+79)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((D_m * M_m) * (D_m * (M_m / d))) / (d * 4.0)) * (0.5 * h)) / l));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 4.5e+79], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.49999999999999994e79

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 69.1

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   5. Applied rewrites 71.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      18. metadata-eval 71.2

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   7. Applied rewrites 71.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}}{4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{4}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \frac{\frac{M}{d} \cdot D}{4}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\color{blue}{\frac{M}{d}} \cdot D\right) \cdot \frac{\frac{M}{d} \cdot D}{4}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\frac{M \cdot D}{d}} \cdot \frac{\frac{M}{d} \cdot D}{4}\right) \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{d \cdot 4}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{d \cdot 4}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}}{d \cdot 4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot M\right)} \cdot \left(\frac{M}{d} \cdot D\right)}{d \cdot 4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\left(D \cdot M\right)} \cdot \left(\frac{M}{d} \cdot D\right)}{d \cdot 4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot M\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{d \cdot 4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot M\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}}{d \cdot 4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot M\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}}{d \cdot 4} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-*.f64 70.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{d \cdot 4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   9. Applied rewrites 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot \frac{M}{d}\right)}{d \cdot 4}} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   if 4.49999999999999994e79 < l

   1. Initial program 56.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 49.8

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      7. lower-sqrt.f64 49.7

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      10. lower-*.f64 49.7

          \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   6. Applied rewrites 49.7%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      7. lower-sqrt.f64 62.8

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 62.8%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 5.4 \cdot 10^{-283}:\\ \;\;\;\;-\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h 5.4e-283)
   (- (/ d (sqrt (* h l))))
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 5.4e-283) {
		tmp = -(d / sqrt((h * l)));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= 5.4d-283) then
        tmp = -(d / sqrt((h * l)))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 5.4e-283) {
		tmp = -(d / Math.sqrt((h * l)));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= 5.4e-283:
		tmp = -(d / math.sqrt((h * l)))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 5.4e-283)
		tmp = Float64(-Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= 5.4e-283)
		tmp = -(d / sqrt((h * l)));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 5.4e-283], (-N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.4e-283

   1. Initial program 66.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 42.1

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      7. lift-pow.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      8. unpow-1 N/A

         \[\leadsto -d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      9. sqrt-div N/A

         \[\leadsto -d \cdot \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \]

      10. metadata-eval N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      11. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. *-commutative N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      15. associate-*r/ N/A

          \[\leadsto -\frac{d \cdot 1}{\sqrt{h \cdot \ell}} \]

      16. *-commutative N/A

          \[\leadsto -\frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      17. *-lft-identity N/A

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

      18. lower-/.f64 42.3

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 42.3%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if 5.4e-283 < h

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 43.6

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      7. lower-sqrt.f64 43.7

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      10. lower-*.f64 43.7

          \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   6. Applied rewrites 43.7%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      7. lower-sqrt.f64 51.3

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 43.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-235}:\\ \;\;\;\;-\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4.8e-235) (- (/ d (sqrt (* h l)))) (* (sqrt (/ (/ 1.0 l) h)) d)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.8e-235) {
		tmp = -(d / sqrt((h * l)));
	} else {
		tmp = sqrt(((1.0 / l) / h)) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-4.8d-235)) then
        tmp = -(d / sqrt((h * l)))
    else
        tmp = sqrt(((1.0d0 / l) / h)) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.8e-235) {
		tmp = -(d / Math.sqrt((h * l)));
	} else {
		tmp = Math.sqrt(((1.0 / l) / h)) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -4.8e-235:
		tmp = -(d / math.sqrt((h * l)))
	else:
		tmp = math.sqrt(((1.0 / l) / h)) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4.8e-235)
		tmp = Float64(-Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -4.8e-235)
		tmp = -(d / sqrt((h * l)));
	else
		tmp = sqrt(((1.0 / l) / h)) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4.8e-235], (-N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.80000000000000022e-235

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 44.9

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   4. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      7. lift-pow.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      8. unpow-1 N/A

         \[\leadsto -d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      9. sqrt-div N/A

         \[\leadsto -d \cdot \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \]

      10. metadata-eval N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      11. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. *-commutative N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      15. associate-*r/ N/A

          \[\leadsto -\frac{d \cdot 1}{\sqrt{h \cdot \ell}} \]

      16. *-commutative N/A

          \[\leadsto -\frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      17. *-lft-identity N/A

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

      18. lower-/.f64 45.2

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 45.2%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -4.80000000000000022e-235 < l

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 41.4

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lower-/.f64 41.4

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. lower-*.f64 41.4

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   6. Applied rewrites 41.4%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      6. lower-/.f64 41.6

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   8. Applied rewrites 41.6%

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 43.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-235}:\\ \;\;\;\;-\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4.8e-235) (- (/ d (sqrt (* h l)))) (* (sqrt (/ 1.0 (* h l))) d)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.8e-235) {
		tmp = -(d / sqrt((h * l)));
	} else {
		tmp = sqrt((1.0 / (h * l))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-4.8d-235)) then
        tmp = -(d / sqrt((h * l)))
    else
        tmp = sqrt((1.0d0 / (h * l))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.8e-235) {
		tmp = -(d / Math.sqrt((h * l)));
	} else {
		tmp = Math.sqrt((1.0 / (h * l))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -4.8e-235:
		tmp = -(d / math.sqrt((h * l)))
	else:
		tmp = math.sqrt((1.0 / (h * l))) * d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4.8e-235)
		tmp = Float64(-Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -4.8e-235)
		tmp = -(d / sqrt((h * l)));
	else
		tmp = sqrt((1.0 / (h * l))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4.8e-235], (-N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.80000000000000022e-235

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 44.9

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   4. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      7. lift-pow.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      8. unpow-1 N/A

         \[\leadsto -d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      9. sqrt-div N/A

         \[\leadsto -d \cdot \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \]

      10. metadata-eval N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      11. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. *-commutative N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      15. associate-*r/ N/A

          \[\leadsto -\frac{d \cdot 1}{\sqrt{h \cdot \ell}} \]

      16. *-commutative N/A

          \[\leadsto -\frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      17. *-lft-identity N/A

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

      18. lower-/.f64 45.2

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 45.2%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -4.80000000000000022e-235 < l

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 41.4

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lower-/.f64 41.4

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. lower-*.f64 41.4

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   6. Applied rewrites 41.4%

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 43.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-235}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))) (if (<= l -4.8e-235) (- t_0) t_0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / sqrt((h * l));
	double tmp;
	if (l <= -4.8e-235) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((h * l))
    if (l <= (-4.8d-235)) then
        tmp = -t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (l <= -4.8e-235) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if l <= -4.8e-235:
		tmp = -t_0
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (l <= -4.8e-235)
		tmp = Float64(-t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (l <= -4.8e-235)
		tmp = -t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.8e-235], (-t$95$0), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if l < -4.80000000000000022e-235

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 44.9

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   4. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      7. lift-pow.f64 N/A

         \[\leadsto -d \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      8. unpow-1 N/A

         \[\leadsto -d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      9. sqrt-div N/A

         \[\leadsto -d \cdot \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \]

      10. metadata-eval N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      11. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. *-commutative N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto -d \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      15. associate-*r/ N/A

          \[\leadsto -\frac{d \cdot 1}{\sqrt{h \cdot \ell}} \]

      16. *-commutative N/A

          \[\leadsto -\frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      17. *-lft-identity N/A

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

      18. lower-/.f64 45.2

          \[\leadsto -\frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 45.2%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -4.80000000000000022e-235 < l

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 41.4

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. unpow-1 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      7. lower-sqrt.f64 41.6

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      10. lower-*.f64 41.6

          \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   6. Applied rewrites 41.6%

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      5. lower-/.f64 41.6

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 41.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 26.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* h l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((h * l));
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((h * l))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((h * l))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. inv-pow N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. lower-pow.f64 N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   6. *-commutative N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   7. lower-*.f64 26.9

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

4. Applied rewrites 26.9%

   \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   3. unpow-1 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. sqrt-div N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. lower-sqrt.f64 26.8

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   10. lower-*.f64 26.8

       \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

6. Applied rewrites 26.8%

   \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   5. lower-/.f64 26.9

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

8. Applied rewrites 26.9%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025104 
(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)))))