Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 77.7%

Time: 19.5s

Alternatives: 24

Speedup: 1.8×

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: 66.2% 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: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\\ t_1 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_2 := 1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_2\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+165}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (pow (/ d l) (pow 2.0 -1.0)))
        (t_1 (* (/ (/ D_m d) 2.0) M_m))
        (t_2 (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l))))
   (if (<= d -2e-310)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) t_2)
     (if (<= d 7.2e+165)
       (* (* (/ (sqrt d) (sqrt h)) t_0) t_2)
       (* (/ (sqrt (pow l -1.0)) (sqrt h)) d)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 = pow((d / l), pow(2.0, -1.0));
	double t_1 = ((D_m / d) / 2.0) * M_m;
	double t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	double tmp;
	if (d <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2;
	} else if (d <= 7.2e+165) {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_2;
	} else {
		tmp = (sqrt(pow(l, -1.0)) / sqrt(h)) * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_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 = (d / l) ** (2.0d0 ** (-1.0d0))
    t_1 = ((d_m / d) / 2.0d0) * m_m
    t_2 = 1.0d0 - ((t_1 * (t_1 * (0.5d0 * h))) / l)
    if (d <= (-2d-310)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2
    else if (d <= 7.2d+165) then
        tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_2
    else
        tmp = (sqrt((l ** (-1.0d0))) / sqrt(h)) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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.pow((d / l), Math.pow(2.0, -1.0));
	double t_1 = ((D_m / d) / 2.0) * M_m;
	double t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	double tmp;
	if (d <= -2e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * t_2;
	} else if (d <= 7.2e+165) {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * t_0) * t_2;
	} else {
		tmp = (Math.sqrt(Math.pow(l, -1.0)) / Math.sqrt(h)) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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.pow((d / l), math.pow(2.0, -1.0))
	t_1 = ((D_m / d) / 2.0) * M_m
	t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l)
	tmp = 0
	if d <= -2e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * t_2
	elif d <= 7.2e+165:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * t_0) * t_2
	else:
		tmp = (math.sqrt(math.pow(l, -1.0)) / math.sqrt(h)) * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 / l) ^ (2.0 ^ -1.0)
	t_1 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_2 = Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * t_2);
	elseif (d <= 7.2e+165)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * t_2);
	else
		tmp = Float64(Float64(sqrt((l ^ -1.0)) / sqrt(h)) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 / l) ^ (2.0 ^ -1.0);
	t_1 = ((D_m / d) / 2.0) * M_m;
	t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	tmp = 0.0;
	if (d <= -2e-310)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2;
	elseif (d <= 7.2e+165)
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_2;
	else
		tmp = (sqrt((l ^ -1.0)) / sqrt(h)) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[d, 7.2e+165], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 74.1

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.1

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

   6. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 72.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 82.8

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

   10. Applied rewrites 82.8%

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

   if -1.999999999999994e-310 < d < 7.1999999999999996e165

   1. Initial program 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 59.5

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 59.5

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 61.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 61.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 74.4

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

   10. Applied rewrites 74.4%

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

   if 7.1999999999999996e165 < d

   1. Initial program 68.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 94.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+165}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{d \cdot \left(4 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-165}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_1 (sqrt (* (/ d l) (/ d h)))))
   (if (<= t_0 -2e-173)
     (*
      (fma
       (* -0.5 (/ (* (* M_m D_m) (* M_m D_m)) (* d (* 4.0 d))))
       (/ h l)
       1.0)
      t_1)
     (if (<= t_0 1e-165)
       (/ d (sqrt (* l h)))
       (if (<= t_0 5e+166)
         (* 1.0 t_1)
         (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_0 <= -2e-173) {
		tmp = fma((-0.5 * (((M_m * D_m) * (M_m * D_m)) / (d * (4.0 * d)))), (h / l), 1.0) * t_1;
	} else if (t_0 <= 1e-165) {
		tmp = d / sqrt((l * h));
	} else if (t_0 <= 5e+166) {
		tmp = 1.0 * t_1;
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (t_0 <= -2e-173)
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(d * Float64(4.0 * d)))), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 1e-165)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (t_0 <= 5e+166)
		tmp = Float64(1.0 * t_1);
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -2e-173], N[(N[(N[(-0.5 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e-165], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+166], N[(1.0 * t$95$1), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.0000000000000001e-173

   1. Initial program 82.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.7

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

   4. Applied rewrites 64.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 63.3

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

   6. Applied rewrites 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 52.1

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. metadata-eval 52.1

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

   8. Applied rewrites 52.1%

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

   if -2.0000000000000001e-173 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e-165

   1. Initial program 67.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   7. Applied rewrites 67.1%

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

   9. Applied rewrites 67.2%

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

   if 1e-165 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e166

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.4

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

   4. Applied rewrites 93.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 93.3%

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

   if 5.0000000000000002e166 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 28.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 37.1%

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

   9. Applied rewrites 37.7%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \left(4 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-165}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+118}:\\ \;\;\;\;t\_0 \cdot \left(\left(\frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell} \cdot -0.125\right) \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{d}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-165}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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) (/ d h))))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_1 -1e+118)
     (* t_0 (* (* (/ (* (* M_m M_m) h) l) -0.125) (/ (/ (* D_m D_m) d) d)))
     (if (<= t_1 1e-165)
       (/ d (sqrt (* l h)))
       (if (<= t_1 5e+166)
         (* 1.0 t_0)
         (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))))

C

D_m = fabs(D);
M_m = fabs(M);
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) * (d / h)));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -1e+118) {
		tmp = t_0 * (((((M_m * M_m) * h) / l) * -0.125) * (((D_m * D_m) / d) / d));
	} else if (t_1 <= 1e-165) {
		tmp = d / sqrt((l * h));
	} else if (t_1 <= 5e+166) {
		tmp = 1.0 * t_0;
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_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 = sqrt(((d / l) * (d / h)))
    t_1 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-1d+118)) then
        tmp = t_0 * (((((m_m * m_m) * h) / l) * (-0.125d0)) * (((d_m * d_m) / d) / d))
    else if (t_1 <= 1d-165) then
        tmp = d / sqrt((l * h))
    else if (t_1 <= 5d+166) then
        tmp = 1.0d0 * t_0
    else
        tmp = ((sqrt(l) * sqrt(h)) ** (-1.0d0)) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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) * (d / h)));
	double t_1 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -1e+118) {
		tmp = t_0 * (((((M_m * M_m) * h) / l) * -0.125) * (((D_m * D_m) / d) / d));
	} else if (t_1 <= 1e-165) {
		tmp = d / Math.sqrt((l * h));
	} else if (t_1 <= 5e+166) {
		tmp = 1.0 * t_0;
	} else {
		tmp = Math.pow((Math.sqrt(l) * Math.sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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) * (d / h)))
	t_1 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -1e+118:
		tmp = t_0 * (((((M_m * M_m) * h) / l) * -0.125) * (((D_m * D_m) / d) / d))
	elif t_1 <= 1e-165:
		tmp = d / math.sqrt((l * h))
	elif t_1 <= 5e+166:
		tmp = 1.0 * t_0
	else:
		tmp = math.pow((math.sqrt(l) * math.sqrt(h)), -1.0) * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(d / l) * Float64(d / h)))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -1e+118)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) / l) * -0.125) * Float64(Float64(Float64(D_m * D_m) / d) / d)));
	elseif (t_1 <= 1e-165)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (t_1 <= 5e+166)
		tmp = Float64(1.0 * t_0);
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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) * (d / h)));
	t_1 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -1e+118)
		tmp = t_0 * (((((M_m * M_m) * h) / l) * -0.125) * (((D_m * D_m) / d) / d));
	elseif (t_1 <= 1e-165)
		tmp = d / sqrt((l * h));
	elseif (t_1 <= 5e+166)
		tmp = 1.0 * t_0;
	else
		tmp = ((sqrt(l) * sqrt(h)) ^ -1.0) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+118], N[(t$95$0 * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-165], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+166], N[(1.0 * t$95$0), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999967e117

   1. Initial program 81.4%

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

   3. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \]

   5. Applied rewrites 57.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.6%

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

   if -9.99999999999999967e117 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e-165

   1. Initial program 76.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 48.9

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

   5. Applied rewrites 48.9%

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 49.0%

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

   if 1e-165 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e166

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.4

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

   4. Applied rewrites 93.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 93.3%

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

   if 5.0000000000000002e166 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 28.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 37.1%

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

   9. Applied rewrites 37.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(\left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-165}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(h \cdot -0.125\right) \cdot \left(\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot \frac{D\_m}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-165}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_1 (sqrt (* (/ d l) (/ d h)))))
   (if (<= t_0 -2e-122)
     (* (* (* h -0.125) (* (* (* M_m M_m) D_m) (/ D_m (* (* d d) l)))) t_1)
     (if (<= t_0 1e-165)
       (/ d (sqrt (* l h)))
       (if (<= t_0 5e+166)
         (* 1.0 t_1)
         (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_0 <= -2e-122) {
		tmp = ((h * -0.125) * (((M_m * M_m) * D_m) * (D_m / ((d * d) * l)))) * t_1;
	} else if (t_0 <= 1e-165) {
		tmp = d / sqrt((l * h));
	} else if (t_0 <= 5e+166) {
		tmp = 1.0 * t_1;
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_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 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt(((d / l) * (d / h)))
    if (t_0 <= (-2d-122)) then
        tmp = ((h * (-0.125d0)) * (((m_m * m_m) * d_m) * (d_m / ((d * d) * l)))) * t_1
    else if (t_0 <= 1d-165) then
        tmp = d / sqrt((l * h))
    else if (t_0 <= 5d+166) then
        tmp = 1.0d0 * t_1
    else
        tmp = ((sqrt(l) * sqrt(h)) ** (-1.0d0)) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_0 <= -2e-122) {
		tmp = ((h * -0.125) * (((M_m * M_m) * D_m) * (D_m / ((d * d) * l)))) * t_1;
	} else if (t_0 <= 1e-165) {
		tmp = d / Math.sqrt((l * h));
	} else if (t_0 <= 5e+166) {
		tmp = 1.0 * t_1;
	} else {
		tmp = Math.pow((Math.sqrt(l) * Math.sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if t_0 <= -2e-122:
		tmp = ((h * -0.125) * (((M_m * M_m) * D_m) * (D_m / ((d * d) * l)))) * t_1
	elif t_0 <= 1e-165:
		tmp = d / math.sqrt((l * h))
	elif t_0 <= 5e+166:
		tmp = 1.0 * t_1
	else:
		tmp = math.pow((math.sqrt(l) * math.sqrt(h)), -1.0) * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (t_0 <= -2e-122)
		tmp = Float64(Float64(Float64(h * -0.125) * Float64(Float64(Float64(M_m * M_m) * D_m) * Float64(D_m / Float64(Float64(d * d) * l)))) * t_1);
	elseif (t_0 <= 1e-165)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (t_0 <= 5e+166)
		tmp = Float64(1.0 * t_1);
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (t_0 <= -2e-122)
		tmp = ((h * -0.125) * (((M_m * M_m) * D_m) * (D_m / ((d * d) * l)))) * t_1;
	elseif (t_0 <= 1e-165)
		tmp = d / sqrt((l * h));
	elseif (t_0 <= 5e+166)
		tmp = 1.0 * t_1;
	else
		tmp = ((sqrt(l) * sqrt(h)) ^ -1.0) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -2e-122], N[(N[(N[(h * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e-165], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+166], N[(1.0 * t$95$1), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000012e-122

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.5

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

   4. Applied rewrites 65.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 64.0

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

   6. Applied rewrites 64.0%

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

   7. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 43.3

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

   9. Applied rewrites 43.3%

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

   if -2.00000000000000012e-122 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e-165

   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. Add Preprocessing

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   7. Applied rewrites 63.7%

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

   9. Applied rewrites 63.7%

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

   if 1e-165 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e166

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.4

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

   4. Applied rewrites 93.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 93.3%

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

   if 5.0000000000000002e166 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 28.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 37.1%

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

   9. Applied rewrites 37.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(h \cdot -0.125\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-165}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(\left(\left(\frac{D\_m}{d} \cdot D\_m\right) \cdot -0.125\right) \cdot M\_m\right), M\_m, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_0 5e+218)
     (*
      (*
       (fma (* -0.5 (pow (/ (* M_m D_m) (* d 2.0)) 2.0)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= t_0 INFINITY)
       (* (/ (sqrt (pow l -1.0)) (sqrt h)) d)
       (fma
        (* (/ (sqrt h) (pow l 1.5)) (* (* (* (/ D_m d) D_m) -0.125) M_m))
        M_m
        (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 5e+218) {
		tmp = (fma((-0.5 * pow(((M_m * D_m) / (d * 2.0)), 2.0)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt(pow(l, -1.0)) / sqrt(h)) * d;
	} else {
		tmp = fma(((sqrt(h) / pow(l, 1.5)) * ((((D_m / d) * D_m) * -0.125) * M_m)), M_m, (d / sqrt((l * h))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 5e+218)
		tmp = Float64(Float64(fma(Float64(-0.5 * (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt((l ^ -1.0)) / sqrt(h)) * d);
	else
		tmp = fma(Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(Float64(Float64(Float64(D_m / d) * D_m) * -0.125) * M_m)), M_m, Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+218], N[(N[(N[(N[(-0.5 * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], N[(N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m + N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999983e218

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.4

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

   6. Applied rewrites 87.4%

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

   if 4.99999999999999983e218 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 53.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 66.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d}} \]

   5. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{D \cdot D}{d} \cdot -0.125\right), \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.4%

      \[\leadsto \mathsf{fma}\left(\left(\left(-0.125 \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot M, \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

   9. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(\left(\left(\frac{D}{d} \cdot D\right) \cdot -0.125\right) \cdot M\right), M, \frac{d}{\sqrt{\ell \cdot h}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(\left(\left(\frac{D}{d} \cdot D\right) \cdot -0.125\right) \cdot M\right), M, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(\left(\left(\frac{D\_m}{d} \cdot D\_m\right) \cdot -0.125\right) \cdot M\_m\right), M\_m, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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_m d) 2.0) M_m))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_1 5e+218)
     (*
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= t_1 INFINITY)
       (* (/ (sqrt (pow l -1.0)) (sqrt h)) d)
       (fma
        (* (/ (sqrt h) (pow l 1.5)) (* (* (* (/ D_m d) D_m) -0.125) M_m))
        M_m
        (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
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_m / d) / 2.0) * M_m;
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 5e+218) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt(pow(l, -1.0)) / sqrt(h)) * d;
	} else {
		tmp = fma(((sqrt(h) / pow(l, 1.5)) * ((((D_m / d) * D_m) * -0.125) * M_m)), M_m, (d / sqrt((l * h))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 5e+218)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt((l ^ -1.0)) / sqrt(h)) * d);
	else
		tmp = fma(Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(Float64(Float64(Float64(D_m / d) * D_m) * -0.125) * M_m)), M_m, Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+218], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], N[(N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m + N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999983e218

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 86.2

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

   6. Applied rewrites 86.8%

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

   if 4.99999999999999983e218 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 53.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 66.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d}} \]

   5. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{D \cdot D}{d} \cdot -0.125\right), \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.4%

      \[\leadsto \mathsf{fma}\left(\left(\left(-0.125 \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot M, \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

   9. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(\left(\left(\frac{D}{d} \cdot D\right) \cdot -0.125\right) \cdot M\right), M, \frac{d}{\sqrt{\ell \cdot h}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(\left(\left(\frac{D}{d} \cdot D\right) \cdot -0.125\right) \cdot M\right), M, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot M\_m, \left(\left(\frac{D\_m}{d} \cdot D\_m\right) \cdot -0.125\right) \cdot M\_m, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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_m d) 2.0) M_m))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l))))))
   (if (<= t_1 5e+218)
     (*
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= t_1 INFINITY)
       (* (/ (sqrt (pow l -1.0)) (sqrt h)) d)
       (fma
        (* (/ (sqrt h) (pow l 1.5)) M_m)
        (* (* (* (/ D_m d) D_m) -0.125) M_m)
        (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
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_m / d) / 2.0) * M_m;
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 5e+218) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt(pow(l, -1.0)) / sqrt(h)) * d;
	} else {
		tmp = fma(((sqrt(h) / pow(l, 1.5)) * M_m), ((((D_m / d) * D_m) * -0.125) * M_m), (d / sqrt((l * h))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 5e+218)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt((l ^ -1.0)) / sqrt(h)) * d);
	else
		tmp = fma(Float64(Float64(sqrt(h) / (l ^ 1.5)) * M_m), Float64(Float64(Float64(Float64(D_m / d) * D_m) * -0.125) * M_m), Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+218], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], N[(N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * M$95$m), $MachinePrecision] + N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999983e218

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 86.2

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

   6. Applied rewrites 86.8%

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

   if 4.99999999999999983e218 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 53.7%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 66.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d}} \]

   5. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot \left(\frac{D \cdot D}{d} \cdot -0.125\right), \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.4%

      \[\leadsto \mathsf{fma}\left(\left(\left(-0.125 \cdot \frac{D \cdot D}{d}\right) \cdot M\right) \cdot M, \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

   9. Applied rewrites 31.5%

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot M, \color{blue}{\left(\left(\frac{D}{d} \cdot D\right) \cdot -0.125\right) \cdot M}, \frac{d}{\sqrt{\ell \cdot h}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot M, \left(\left(\frac{D}{d} \cdot D\right) \cdot -0.125\right) \cdot M, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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_m d) 2.0) M_m)))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
        5e+218)
     (*
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (* (/ (sqrt (pow l -1.0)) (sqrt h)) d))))

C

D_m = fabs(D);
M_m = fabs(M);
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_m / d) / 2.0) * M_m;
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+218) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = (sqrt(pow(l, -1.0)) / sqrt(h)) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(D_m / d) / 2.0) * M_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+218)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(sqrt((l ^ -1.0)) / sqrt(h)) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+218], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999983e218

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 86.2

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

   6. Applied rewrites 86.8%

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

   if 4.99999999999999983e218 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 25.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.2

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

   5. Applied rewrites 37.2%

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

   7. Applied rewrites 38.0%

      \[\leadsto \frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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_m d) 2.0) M_m)))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
        5e+218)
     (*
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))

C

D_m = fabs(D);
M_m = fabs(M);
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_m / d) / 2.0) * M_m;
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+218) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(D_m / d) / 2.0) * M_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+218)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+218], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999983e218

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 86.2

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

   6. Applied rewrites 86.8%

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

   if 4.99999999999999983e218 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 25.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.2

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

   5. Applied rewrites 37.2%

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

   7. Applied rewrites 37.2%

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

   9. Applied rewrites 37.8%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot \frac{D\_m}{4 \cdot d}\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
      5e+166)
   (*
    (fma (* -0.5 (/ (* (* M_m D_m) (* M_m (/ D_m (* 4.0 d)))) d)) (/ h l) 1.0)
    (sqrt (* (/ d l) (/ d h))))
   (* (pow (* (sqrt l) (sqrt h)) -1.0) d)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+166) {
		tmp = fma((-0.5 * (((M_m * D_m) * (M_m * (D_m / (4.0 * d)))) / d)), (h / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+166)
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * Float64(D_m / Float64(4.0 * d)))) / d)), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+166], N[(N[(N[(-0.5 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * N[(D$95$m / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e166

   1. Initial program 87.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.2

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

   4. Applied rewrites 71.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 70.7

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

   6. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 70.7%

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

   if 5.0000000000000002e166 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 28.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 37.1%

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

   9. Applied rewrites 37.7%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot \frac{D}{4 \cdot d}\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ \mathbf{if}\;d \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+165}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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_m d) 2.0) M_m)))
   (if (<= d 4.3e-308)
     (*
      (*
       (fma (* -0.5 (pow (/ (* M_m D_m) (* d 2.0)) 2.0)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (if (<= d 7.2e+165)
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0)))
        (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
       (* (/ (sqrt (pow l -1.0)) (sqrt h)) d)))))

C

D_m = fabs(D);
M_m = fabs(M);
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_m / d) / 2.0) * M_m;
	double tmp;
	if (d <= 4.3e-308) {
		tmp = (fma((-0.5 * pow(((M_m * D_m) / (d * 2.0)), 2.0)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= 7.2e+165) {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else {
		tmp = (sqrt(pow(l, -1.0)) / sqrt(h)) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(D_m / d) / 2.0) * M_m)
	tmp = 0.0
	if (d <= 4.3e-308)
		tmp = Float64(Float64(fma(Float64(-0.5 * (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= 7.2e+165)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l)));
	else
		tmp = Float64(Float64(sqrt((l ^ -1.0)) / sqrt(h)) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[d, 4.3e-308], N[(N[(N[(N[(-0.5 * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e+165], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 4.3000000000000002e-308

   1. Initial program 71.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if 4.3000000000000002e-308 < d < 7.1999999999999996e165

   1. Initial program 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 60.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 61.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 61.9

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 75.1

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

   10. Applied rewrites 75.1%

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

   if 7.1999999999999996e165 < d

   1. Initial program 68.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 94.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+165}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot M\_m\\ t_1 := \left(\left(-h\right) \cdot \left(\frac{0.125}{\ell} \cdot \left(t\_0 \cdot t\_0\right) - {h}^{-1}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq -1.08 \cdot 10^{-244}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-87}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\frac{M\_m \cdot M\_m}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+167}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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_m d) M_m))
        (t_1
         (*
          (* (- h) (- (* (/ 0.125 l) (* t_0 t_0)) (pow h -1.0)))
          (sqrt (* (/ d l) (/ d h))))))
   (if (<= d -1.08e-244)
     t_1
     (if (<= d 1.05e-87)
       (*
        (*
         (fma (* (* -0.125 (* D_m D_m)) (/ (/ (* M_m M_m) d) d)) (/ h l) 1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (if (<= d 6.8e+167) t_1 (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
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_m / d) * M_m;
	double t_1 = (-h * (((0.125 / l) * (t_0 * t_0)) - pow(h, -1.0))) * sqrt(((d / l) * (d / h)));
	double tmp;
	if (d <= -1.08e-244) {
		tmp = t_1;
	} else if (d <= 1.05e-87) {
		tmp = (fma(((-0.125 * (D_m * D_m)) * (((M_m * M_m) / d) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (d <= 6.8e+167) {
		tmp = t_1;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(D_m / d) * M_m)
	t_1 = Float64(Float64(Float64(-h) * Float64(Float64(Float64(0.125 / l) * Float64(t_0 * t_0)) - (h ^ -1.0))) * sqrt(Float64(Float64(d / l) * Float64(d / h))))
	tmp = 0.0
	if (d <= -1.08e-244)
		tmp = t_1;
	elseif (d <= 1.05e-87)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(Float64(M_m * M_m) / d) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (d <= 6.8e+167)
		tmp = t_1;
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-h) * N[(N[(N[(0.125 / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.08e-244], t$95$1, If[LessEqual[d, 1.05e-87], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.8e+167], t$95$1, N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.07999999999999996e-244 or 1.05000000000000004e-87 < d < 6.8000000000000001e167

   1. Initial program 76.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 65.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 64.6

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

   6. Applied rewrites 64.6%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{0.125}{\ell} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) - \frac{1}{h}\right)\right)} \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -1.07999999999999996e-244 < d < 1.05000000000000004e-87

   1. Initial program 44.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 44.8%

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

   5. Taylor expanded in d around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 37.4

          \[\leadsto \left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Applied rewrites 37.4%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 6.8000000000000001e167 < d

   1. Initial program 67.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 79.3%

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

   9. Applied rewrites 79.3%

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

   11. Applied rewrites 94.4%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.08 \cdot 10^{-244}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{0.125}{\ell} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) - {h}^{-1}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-87}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+167}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{0.125}{\ell} \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) - {h}^{-1}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 7.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{\frac{D\_m}{d}}{2} \cdot M\_m\right)}^{2}, 1\right) \cdot t\_0\right)}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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))))
   (if (<= d 4.3e-308)
     (*
      (*
       (fma (* -0.5 (pow (/ (* M_m D_m) (* d 2.0)) 2.0)) (/ h l) 1.0)
       (sqrt (/ d h)))
      t_0)
     (if (<= d 7.1e+165)
       (/
        (*
         (sqrt d)
         (*
          (fma (* (/ h l) -0.5) (pow (* (/ (/ D_m d) 2.0) M_m) 2.0) 1.0)
          t_0))
        (sqrt h))
       (* (/ (sqrt (pow l -1.0)) (sqrt h)) d)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 tmp;
	if (d <= 4.3e-308) {
		tmp = (fma((-0.5 * pow(((M_m * D_m) / (d * 2.0)), 2.0)), (h / l), 1.0) * sqrt((d / h))) * t_0;
	} else if (d <= 7.1e+165) {
		tmp = (sqrt(d) * (fma(((h / l) * -0.5), pow((((D_m / d) / 2.0) * M_m), 2.0), 1.0) * t_0)) / sqrt(h);
	} else {
		tmp = (sqrt(pow(l, -1.0)) / sqrt(h)) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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))
	tmp = 0.0
	if (d <= 4.3e-308)
		tmp = Float64(Float64(fma(Float64(-0.5 * (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * t_0);
	elseif (d <= 7.1e+165)
		tmp = Float64(Float64(sqrt(d) * Float64(fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(Float64(D_m / d) / 2.0) * M_m) ^ 2.0), 1.0) * t_0)) / sqrt(h));
	else
		tmp = Float64(Float64(sqrt((l ^ -1.0)) / sqrt(h)) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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]}, If[LessEqual[d, 4.3e-308], N[(N[(N[(N[(-0.5 * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 7.1e+165], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 4.3000000000000002e-308

   1. Initial program 71.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if 4.3000000000000002e-308 < d < 7.09999999999999976e165

   1. Initial program 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 58.9%

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

   6. Applied rewrites 71.4%

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

   if 7.09999999999999976e165 < d

   1. Initial program 68.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 77.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.6%

      \[\leadsto \frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;d \leq -1.04 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{t\_0 \cdot t\_0}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\frac{M\_m \cdot M\_m}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= d -1.04e-244)
     (*
      (fma (* -0.5 (/ (* t_0 t_0) 4.0)) (/ h l) 1.0)
      (sqrt (* (/ d l) (/ d h))))
     (if (<= d 4.6e+64)
       (*
        (*
         (fma (* (* -0.125 (* D_m D_m)) (/ (/ (* M_m M_m) d) d)) (/ h l) 1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (* (pow (* (sqrt l) (sqrt h)) -1.0) d)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (d <= -1.04e-244) {
		tmp = fma((-0.5 * ((t_0 * t_0) / 4.0)), (h / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (d <= 4.6e+64) {
		tmp = (fma(((-0.125 * (D_m * D_m)) * (((M_m * M_m) / d) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (d <= -1.04e-244)
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(t_0 * t_0) / 4.0)), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= 4.6e+64)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(Float64(M_m * M_m) / d) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[d, -1.04e-244], N[(N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+64], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.04000000000000003e-244

   1. Initial program 75.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 75.5

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{2 \cdot 2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{2 \cdot 2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{2 \cdot 2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. metadata-eval 64.7

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Applied rewrites 64.7%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -1.04000000000000003e-244 < d < 4.6e64

   1. Initial program 57.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-/r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-*.f64 46.2

          \[\leadsto \left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Applied rewrites 46.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 4.6e64 < d

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 67.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 84.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.04 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.04 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\frac{M\_m \cdot D\_m}{d} \cdot \left(D\_m \cdot M\_m\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\frac{M\_m \cdot M\_m}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= d -1.04e-244)
   (*
    (fma
     (* -0.5 (/ (* (/ (* M_m D_m) d) (* D_m M_m)) (* 2.0 (* d 2.0))))
     (/ h l)
     1.0)
    (sqrt (* (/ d l) (/ d h))))
   (if (<= d 4.6e+64)
     (*
      (*
       (fma (* (* -0.125 (* D_m D_m)) (/ (/ (* M_m M_m) d) d)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (d <= -1.04e-244) {
		tmp = fma((-0.5 * ((((M_m * D_m) / d) * (D_m * M_m)) / (2.0 * (d * 2.0)))), (h / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (d <= 4.6e+64) {
		tmp = (fma(((-0.125 * (D_m * D_m)) * (((M_m * M_m) / d) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (d <= -1.04e-244)
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(Float64(M_m * D_m) / d) * Float64(D_m * M_m)) / Float64(2.0 * Float64(d * 2.0)))), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= 4.6e+64)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(Float64(M_m * M_m) / d) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[d, -1.04e-244], N[(N[(N[(-0.5 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+64], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.04000000000000003e-244

   1. Initial program 75.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 75.5

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      22. lift-*.f64 63.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Applied rewrites 63.6%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\color{blue}{\frac{M}{d}} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\frac{M \cdot D}{d}} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\frac{\color{blue}{D \cdot M}}{d} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\frac{\color{blue}{D \cdot M}}{d} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lower-/.f64 63.7

         \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\color{blue}{\frac{D \cdot M}{d}} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\frac{\color{blue}{D \cdot M}}{d} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\frac{\color{blue}{M \cdot D}}{d} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lower-*.f64 63.7

         \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\frac{\color{blue}{M \cdot D}}{d} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Applied rewrites 63.7%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\color{blue}{\frac{M \cdot D}{d}} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -1.04000000000000003e-244 < d < 4.6e64

   1. Initial program 57.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-/r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-*.f64 46.2

          \[\leadsto \left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Applied rewrites 46.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 4.6e64 < d

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 67.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 84.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.04 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\frac{M \cdot D}{d} \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.04 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\frac{M\_m \cdot M\_m}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= d -1.04e-244)
   (*
    (fma
     (* -0.5 (/ (* (* (/ M_m d) D_m) (* D_m M_m)) (* 2.0 (* d 2.0))))
     (/ h l)
     1.0)
    (sqrt (* (/ d l) (/ d h))))
   (if (<= d 4.6e+64)
     (*
      (*
       (fma (* (* -0.125 (* D_m D_m)) (/ (/ (* M_m M_m) d) d)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (d <= -1.04e-244) {
		tmp = fma((-0.5 * ((((M_m / d) * D_m) * (D_m * M_m)) / (2.0 * (d * 2.0)))), (h / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (d <= 4.6e+64) {
		tmp = (fma(((-0.125 * (D_m * D_m)) * (((M_m * M_m) / d) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (d <= -1.04e-244)
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(Float64(M_m / d) * D_m) * Float64(D_m * M_m)) / Float64(2.0 * Float64(d * 2.0)))), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= 4.6e+64)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(Float64(M_m * M_m) / d) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[d, -1.04e-244], N[(N[(N[(-0.5 * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+64], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.04000000000000003e-244

   1. Initial program 75.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 75.5

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      22. lift-*.f64 63.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Applied rewrites 63.6%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -1.04000000000000003e-244 < d < 4.6e64

   1. Initial program 57.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-/r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-*.f64 46.2

          \[\leadsto \left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Applied rewrites 46.2%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 4.6e64 < d

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 67.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 84.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.04 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot \frac{D\_m}{4 \cdot d}\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\frac{M\_m \cdot M\_m}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= d -3.1e-11)
   (*
    (fma (* -0.5 (/ (* (* M_m D_m) (* M_m (/ D_m (* 4.0 d)))) d)) (/ h l) 1.0)
    (sqrt (* (/ d l) (/ d h))))
   (if (<= d 4.6e+64)
     (*
      (*
       (fma (* (* -0.125 (* D_m D_m)) (/ (/ (* M_m M_m) d) d)) (/ h l) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (* (pow (* (sqrt l) (sqrt h)) -1.0) d))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (d <= -3.1e-11) {
		tmp = fma((-0.5 * (((M_m * D_m) * (M_m * (D_m / (4.0 * d)))) / d)), (h / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (d <= 4.6e+64) {
		tmp = (fma(((-0.125 * (D_m * D_m)) * (((M_m * M_m) / d) / d)), (h / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (d <= -3.1e-11)
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * Float64(D_m / Float64(4.0 * d)))) / d)), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= 4.6e+64)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(Float64(M_m * M_m) / d) / d)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[d, -3.1e-11], N[(N[(N[(-0.5 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * N[(D$95$m / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+64], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.10000000000000028e-11

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 78.0

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   4. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      22. lift-*.f64 67.1

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Applied rewrites 67.1%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{M}{d}} \cdot D\right) \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{M \cdot D}{d}} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{d} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{\color{blue}{D \cdot M}}{d} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot M\right) \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot M\right) \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Applied rewrites 67.2%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(M \cdot \frac{D}{4 \cdot d}\right)}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -3.10000000000000028e-11 < d < 4.6e64

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-/r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-*.f64 48.5

          \[\leadsto \left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Applied rewrites 48.5%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 4.6e64 < d

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 67.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 84.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot \frac{D}{4 \cdot d}\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.4 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= d 6.4e+94)
   (*
    (fma
     (* -0.5 (/ (* (* (/ M_m d) D_m) (* D_m M_m)) (* 2.0 (* d 2.0))))
     (/ h l)
     1.0)
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (* (pow (* (sqrt l) (sqrt h)) -1.0) d)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (d <= 6.4e+94) {
		tmp = fma((-0.5 * ((((M_m / d) * D_m) * (D_m * M_m)) / (2.0 * (d * 2.0)))), (h / l), 1.0) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (d <= 6.4e+94)
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(Float64(M_m / d) * D_m) * Float64(D_m * M_m)) / Float64(2.0 * Float64(d * 2.0)))), Float64(h / l), 1.0) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[d, 6.4e+94], N[(N[(N[(-0.5 * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.40000000000000028e94

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 66.1

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   4. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      22. lift-*.f64 52.2

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Applied rewrites 52.2%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \color{blue}{{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}^{\frac{1}{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\frac{1}{2}} \]

      4. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \]

      11. lower-*.f64 65.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \]

      13. metadata-eval 65.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \]

      15. unpow1/2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \]

      16. lower-sqrt.f64 65.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \]

      18. metadata-eval 65.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \]

      19. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \]

      20. unpow1/2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      21. lower-sqrt.f64 65.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

   8. Applied rewrites 65.6%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

   if 6.40000000000000028e94 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 69.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 84.8%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.4 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.4 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(D\_m \cdot \left(\frac{M\_m}{d} \cdot \left(M\_m \cdot \frac{D\_m}{4 \cdot d}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= d 6.4e+94)
   (*
    (fma (* -0.5 (* D_m (* (/ M_m d) (* M_m (/ D_m (* 4.0 d)))))) (/ h l) 1.0)
    (sqrt (* (/ d l) (/ d h))))
   (* (pow (* (sqrt l) (sqrt h)) -1.0) d)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (d <= 6.4e+94) {
		tmp = fma((-0.5 * (D_m * ((M_m / d) * (M_m * (D_m / (4.0 * d)))))), (h / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (d <= 6.4e+94)
		tmp = Float64(fma(Float64(-0.5 * Float64(D_m * Float64(Float64(M_m / d) * Float64(M_m * Float64(D_m / Float64(4.0 * d)))))), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[d, 6.4e+94], N[(N[(N[(-0.5 * N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m * N[(D$95$m / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.40000000000000028e94

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 66.1

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   4. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      22. lift-*.f64 52.2

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Applied rewrites 52.2%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}}{2 \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(D \cdot \frac{M}{d}\right)} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{D \cdot M}{2 \cdot \left(d \cdot 2\right)}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{\color{blue}{D \cdot M}}{2 \cdot \left(d \cdot 2\right)}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot \left(d \cdot 2\right)}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot \left(d \cdot 2\right)}\right)}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot \left(d \cdot 2\right)}\right)}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-/.f64 50.7

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \color{blue}{\frac{D}{2 \cdot \left(d \cdot 2\right)}}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{\color{blue}{2 \cdot \left(d \cdot 2\right)}}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{2 \cdot \color{blue}{\left(2 \cdot d\right)}}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{\color{blue}{\left(2 \cdot 2\right) \cdot d}}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{\color{blue}{\left(2 \cdot 2\right) \cdot d}}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. metadata-eval 50.7

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{\color{blue}{4} \cdot d}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   8. Applied rewrites 50.7%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{4 \cdot d}\right)\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if 6.40000000000000028e94 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 69.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 84.8%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.4 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{D}{4 \cdot d}\right)\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{-296}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -1.45e+49)
   (* 1.0 (sqrt (* (/ d l) (/ d h))))
   (if (<= h 2.4e-296)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -1.45e+49) {
		tmp = 1.0 * sqrt(((d / l) * (d / h)));
	} else if (h <= 2.4e-296) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m =     private
M_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 <= (-1.45d+49)) then
        tmp = 1.0d0 * sqrt(((d / l) * (d / h)))
    else if (h <= 2.4d-296) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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 <= -1.45e+49) {
		tmp = 1.0 * Math.sqrt(((d / l) * (d / h)));
	} else if (h <= 2.4e-296) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 <= -1.45e+49:
		tmp = 1.0 * math.sqrt(((d / l) * (d / h)))
	elif h <= 2.4e-296:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -1.45e+49)
		tmp = Float64(1.0 * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (h <= 2.4e-296)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 <= -1.45e+49)
		tmp = 1.0 * sqrt(((d / l) * (d / h)));
	elseif (h <= 2.4e-296)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -1.45e+49], N[(1.0 * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.4e-296], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.45e49

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lower-*.f64 64.5

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   4. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      9. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      12. frac-times N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(M \cdot D\right)}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot \left(2 \cdot d\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      22. lift-*.f64 54.3

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \color{blue}{\left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   6. Applied rewrites 54.3%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot M\right)}{2 \cdot \left(d \cdot 2\right)}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   7. Taylor expanded in d around inf

      \[\leadsto \color{blue}{1} \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.2%

      \[\leadsto \color{blue}{1} \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

   if -1.45e49 < h < 2.39999999999999996e-296

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{-1}\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-1 \cdot d\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      11. *-commutative N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      12. lower-*.f64 58.4

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 2.39999999999999996e-296 < h

   1. Initial program 60.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 41.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.1%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 51.4%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{-296}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 46.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-296}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 2.4e-296)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= 2.4e-296) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m =     private
M_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 <= 2.4d-296) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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 <= 2.4e-296) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 <= 2.4e-296:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 <= 2.4e-296)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 <= 2.4e-296)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, 2.4e-296], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 2.39999999999999996e-296

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{-1}\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-1 \cdot d\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      11. *-commutative N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      12. lower-*.f64 41.4

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 2.39999999999999996e-296 < h

   1. Initial program 60.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 41.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.1%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 51.4%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.4 \cdot 10^{-296}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{-179}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (pow (* l h) -1.0))))
   (if (<= d -5.8e-179) (* (- d) t_0) (* t_0 d))))

C

D_m = fabs(D);
M_m = fabs(M);
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(pow((l * h), -1.0));
	double tmp;
	if (d <= -5.8e-179) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_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 = sqrt(((l * h) ** (-1.0d0)))
    if (d <= (-5.8d-179)) then
        tmp = -d * t_0
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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(Math.pow((l * h), -1.0));
	double tmp;
	if (d <= -5.8e-179) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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(math.pow((l * h), -1.0))
	tmp = 0
	if d <= -5.8e-179:
		tmp = -d * t_0
	else:
		tmp = t_0 * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(l * h) ^ -1.0))
	tmp = 0.0
	if (d <= -5.8e-179)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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(((l * h) ^ -1.0));
	tmp = 0.0;
	if (d <= -5.8e-179)
		tmp = -d * t_0;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.8e-179], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.7999999999999998e-179

   1. Initial program 76.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{-1}\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-1 \cdot d\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      11. *-commutative N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      12. lower-*.f64 49.0

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -5.7999999999999998e-179 < d

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 37.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-179}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 26.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (* (sqrt (pow (* l h) -1.0)) d))

C

D_m = fabs(D);
M_m = fabs(M);
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 sqrt(pow((l * h), -1.0)) * d;
}

Fortran

D_m =     private
M_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 = sqrt(((l * h) ** (-1.0d0))) * d
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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 Math.sqrt(Math.pow((l * h), -1.0)) * d;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 math.sqrt(math.pow((l * h), -1.0)) * d

Julia

D_m = abs(D)
M_m = abs(M)
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(sqrt((Float64(l * h) ^ -1.0)) * d)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 = sqrt(((l * h) ^ -1.0)) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 27.3

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 27.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

6. Final simplification 27.3%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
7. Add Preprocessing

Alternative 24: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (* l h))))

C

D_m = fabs(D);
M_m = fabs(M);
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((l * h));
}

Fortran

D_m =     private
M_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((l * h))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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((l * h));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
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(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 27.3

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 27.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 27.3%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
8. Step-by-step derivation

9. Applied rewrites 27.3%

   \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024357 
(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)))))